text:
# Mecca Compute The Differential And Tangent Vector Of Vector Function Pdf

## Vector ﬁelds UPMC

### Partial derivative. Total differential. Total derivative.

Sage Reference Manual Differential Geometry of. Likewise, in differential geometry, the differential of a function at a point is a linear function of a tangent vector (an "infinitely small displacement"), which exhibits it as a kind of one-form: the exterior derivative of the function., 2011/01/23 · Determining Curvature of a Curve Defined by a Vector Valued Function Mathispower4u Loading... Unsubscribe from Mathispower4u? Cancel ….

### Differential of a function Wikipedia

13.2 Calculus with vector functions. use the unit tangent vector ~tand a stepsize hto compute x 1 = x 0 + h~t and then apply Newton’s method to determine a point x 2 which satis es the implicit function, and has the same value of the p-coordinate as x 1. To do all, Likewise, in differential geometry, the differential of a function at a point is a linear function of a tangent vector (an "infinitely small displacement"), which exhibits it as a kind of one-form: the exterior derivative of the function..

Engineering Mathematics - I Semester – 1 By Dr N V Nagendram UNIT – V Vector Differential Calculus Gradient, Divergence and Curl 1.01 Introduction 1.02 Vector Differentiation 1.03 Problems Exercise and solutions 1.04 2018/05/30 · In this section we will compute the differential for a function. We will give an application of differentials in this section. However, one of the more important uses of differentials will come in the next chapter and unfortunately we will not

VECTOR DIFFERENTIATION Differentiation of vector functions. In calculus we compute derivatives of real functions of a real variable. In the case of functions of a single variable y = f(x) we compute the derivative of y with 2018/05/30 · In this section we will compute the differential for a function. We will give an application of differentials in this section. However, one of the more important uses of differentials will come in the next chapter and unfortunately we will not

1.1. CURVES 2 lines), acceleration, and jerk that completely determine the path of the curve when we use some parameter t to travel along it. q v a j Tangent Line Most of the curves we study will be given as parametrized curves,i.e., use the unit tangent vector ~tand a stepsize hto compute x 1 = x 0 + h~t and then apply Newton’s method to determine a point x 2 which satis es the implicit function, and has the same value of the p-coordinate as x 1. To do all

More generally, if v is any vector in Rm, then the product D pf(v) is called the directional derivative of f in the direction of v. This is something like a \partial derivative" in the direction of the vector v. The directional derivative D p(v) can be On other occasions it will be useful to work with a unit vector in the same direction as ${\bf r}'$; of course, we can compute such a vector by dividing ${\bf r}'$ by its own length. This standard unit tangent vector is usually denoted by

Chapter 4 Diﬀerentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f: D → R, where D is a subset of Rn, where n is the number of variables. These In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can

• “Discrete Differential‐Geometry Operators for Triangulated 2‐ Manifolds”, Meyer et al., ’02 • “Restricted Delaunay triangulations and normal cycle”, Cohen‐Steiner et al., SoCG ‘03 • “On the convergence of metric and geometric 2013/09/23 · Compute unit normal vector, unit tangent vector, and curvature. Description of basic geometry and an example. Compute unit normal vector, unit tangent vector, and curvature. Description of basic geometry and an example. Sign in

The function F always takes two arguments, the scalar independent variable, t, and the vector of dependent variables, y. A program that evaluates F(t;y) hould compute the derivatives of all the state variables and return them in On other occasions it will be useful to work with a unit vector in the same direction as ${\bf r}'$; of course, we can compute such a vector by dividing ${\bf r}'$ by its own length. This standard unit tangent vector is usually denoted by

Concrete example of the derivative of a vector valued function to better understand what it means If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter Concrete example of the derivative of a vector valued function to better understand what it means If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter

What is a covector and what is it used for? Ask Question Asked 6 years, 11 months ago Active 1 year, 10 months ago Viewed 23k times 48 69 $\begingroup$ From what I understand, a covector is an object that takes a vector 2 Basic di erential geometry Conventions If UˆRm is open, V is a real (or complex) vector space (of nite dimension), and ’: U!V is a smooth function, then the partial derivative of ’with respect to x i is denoted in the following di erent ways,

2011/01/23 · Determining Curvature of a Curve Defined by a Vector Valued Function Mathispower4u Loading... Unsubscribe from Mathispower4u? Cancel … Section 14.5, Directional derivatives and gradient vectors p. 331 (3/23/08) Estimating directional derivatives from level curves We could ﬁnd approximate values of directional derivatives from level curves by using the techniques of the

meaning? We say that a parametric curve and c: (a, b) —Y is footed at p max. flowline for V footed at p. VF V on Rn, maximal it extends other any A flowline for V footed at p is said to be V be a VF on Let Let flowline for V footed at 2018/05/31 · Section 6-7 : Calculus with Vector Functions In this section we need to talk briefly about limits, derivatives and integrals of vector functions. As you will see, these behave in a fairly predictable manner. We will be doing all of the work

Likewise, in differential geometry, the differential of a function at a point is a linear function of a tangent vector (an "infinitely small displacement"), which exhibits it as a kind of one-form: the exterior derivative of the function. 2.3 Binormal vector and torsion Figure 2.6: The tangent, normal, and binormal vectors define an orthogonal coordinate system along a space curve In Sects. 2.1 and 2.2 , we have introduced the tangent and normal vectors, which are orthogonal to each other and lie in the osculating plane.

2 CHAPTER 1. WHAT IS DIFFERENTIAL GEOMETRY? U f Figure 1.1: A chart Perhaps the user of such a map will be content to use the map to plot the shortest path between two points pand qin U. This path is called a geodesic. More generally, if v is any vector in Rm, then the product D pf(v) is called the directional derivative of f in the direction of v. This is something like a \partial derivative" in the direction of the vector v. The directional derivative D p(v) can be

Engineering Mathematics - I Semester – 1 By Dr N V Nagendram UNIT – V Vector Differential Calculus Gradient, Divergence and Curl 1.01 Introduction 1.02 Vector Differentiation 1.03 Problems Exercise and solutions 1.04 Engineering Mathematics - I Semester – 1 By Dr N V Nagendram UNIT – V Vector Differential Calculus Gradient, Divergence and Curl 1.01 Introduction 1.02 Vector Differentiation 1.03 Problems Exercise and solutions 1.04

3 III. Chain Rule, change of variables and implicit functions. { How to compute the total diﬁerential for the vector function of a vector vari-able? { What is a Jacobi matrix of a map f: Rn! Rm? { What is the inverse map f¡1 for the map f: Rn! Engineering Mathematics - I Semester – 1 By Dr N V Nagendram UNIT – V Vector Differential Calculus Gradient, Divergence and Curl 1.01 Introduction 1.02 Vector Differentiation 1.03 Problems Exercise and solutions 1.04

A Euclidean vector field V for which each vector V(p) is tangent to M at p is called a tangent vector field on M (Fig. 4.25). Frequently these vector fields are defined, not on all of M, but only on some region in M. As usual, we Sage Reference Manual: Differential Geometry of Curves and Surfaces, Release 8.9 (continued from previous page) sage: # Find minimum and max of the gauss curvature

### (Discrete) Differential Geometry

Section 14.5 (3/23/08) Directional derivatives and. 264 CHAPTER 9. CALCULUS ON EUCLIDEAN SPACE Deﬁnition 9.1. A tangent vector vp to Euclidean space Rn consists of a pair of elements v,p of Rn; v is called the vector part and p is called the point of application of vp. v v p, 3 III. Chain Rule, change of variables and implicit functions. { How to compute the total diﬁerential for the vector function of a vector vari-able? { What is a Jacobi matrix of a map f: Rn! Rm? { What is the inverse map f¡1 for the map f: Rn!.

Calculus and Diﬀerential Geometry An Introduction. On other occasions it will be useful to work with a unit vector in the same direction as ${\bf r}'$; of course, we can compute such a vector by dividing ${\bf r}'$ by its own length. This standard unit tangent vector is usually denoted by, In vector (or multivariable) calculus, we will deal with functions of two or three variables (usually x,y or x,y,z, respectively). The graph of a function of two variables, say, z=f(x,y), lies in Euclidean space, which in the Cartesian a,b,c)..

### An Operator Approach to Tangent Vector Field

Classical Differential Geometry. Section 14.5, Directional derivatives and gradient vectors p. 331 (3/23/08) Estimating directional derivatives from level curves We could ﬁnd approximate values of directional derivatives from level curves by using the techniques of the Home Calculators Calculus III Calculators Math Problem Solver (all calculators) Unit Tangent Vector Calculator The calculator will find the unit tangent vector of a vector-valued function at the given point, with steps shown..

• (Discrete) Differential Geometry
• Calculus I Differentials

• use the unit tangent vector ~tand a stepsize hto compute x 1 = x 0 + h~t and then apply Newton’s method to determine a point x 2 which satis es the implicit function, and has the same value of the p-coordinate as x 1. To do all 264 CHAPTER 9. CALCULUS ON EUCLIDEAN SPACE Deﬁnition 9.1. A tangent vector vp to Euclidean space Rn consists of a pair of elements v,p of Rn; v is called the vector part and p is called the point of application of vp. v v p

NOTES ON DIFFERENTIAL GEOMETRY 3 the ﬁrst derivative of x: (6) t = dx/ds = x˙ Note that this is a unit vector precisely because we have assumed that the parameterization of the curve is unit-speed. The second derivative ¨x will O. Azencot & M. Ben-Chen & F. Chazal & M. Ovsjanikov / An Operator Approach to Tangent Vector Field Processing An alternative approach in the continuous case, is to work with differential forms (see e.g. [Mor01]) which are linear

1.1 VECTOR CALCULUS (Derivative of Scalar & Vector Function) - Free download as Powerpoint Presentation (.ppt), PDF File (.pdf), Text File (.txt) or view presentation slides online. k m8 k m8 Search Search In vector (or multivariable) calculus, we will deal with functions of two or three variables (usually x,y or x,y,z, respectively). The graph of a function of two variables, say, z=f(x,y), lies in Euclidean space, which in the Cartesian a,b,c).

Differential Calculus of Vector Valued Functions Functions of Several Variables We are going to consider scalar valued and vector valued functions of several real variables. For example, z f x,y , w F x,y,z , y G x1,x2,...,xn V v 1 x,y 1.1. CURVES 2 lines), acceleration, and jerk that completely determine the path of the curve when we use some parameter t to travel along it. q v a j Tangent Line Most of the curves we study will be given as parametrized curves,i.e.,

2018/05/30 · In this section we will compute the differential for a function. We will give an application of differentials in this section. However, one of the more important uses of differentials will come in the next chapter and unfortunately we will not 1.1. CURVES 2 lines), acceleration, and jerk that completely determine the path of the curve when we use some parameter t to travel along it. q v a j Tangent Line Most of the curves we study will be given as parametrized curves,i.e.,

More generally, if v is any vector in Rm, then the product D pf(v) is called the directional derivative of f in the direction of v. This is something like a \partial derivative" in the direction of the vector v. The directional derivative D p(v) can be Barrett O'Neill, in Elementary Differential Geometry (Second Edition), 2006 1.3 Directional Derivatives Associated with each tangent vector v p to R 3 is the straight line t → p + tv (see Example 4.2). If f is a differentiable function on R 3, then

2011/01/23 · Determining Curvature of a Curve Defined by a Vector Valued Function Mathispower4u Loading... Unsubscribe from Mathispower4u? Cancel … What is a covector and what is it used for? Ask Question Asked 6 years, 11 months ago Active 1 year, 10 months ago Viewed 23k times 48 69 $\begingroup$ From what I understand, a covector is an object that takes a vector

Vector Calculator Solve vector operations and functions step-by-step Matrices Add, Subtract Multiply, Power Trace Transpose Determinant Inverse Rank Minors & Cofactors Characteristic Polynomial Gauss Jordan (RREF) The function F always takes two arguments, the scalar independent variable, t, and the vector of dependent variables, y. A program that evaluates F(t;y) hould compute the derivatives of all the state variables and return them in

2.3 Binormal vector and torsion Figure 2.6: The tangent, normal, and binormal vectors define an orthogonal coordinate system along a space curve In Sects. 2.1 and 2.2 , we have introduced the tangent and normal vectors, which are orthogonal to each other and lie in the osculating plane. 2.3 Binormal vector and torsion Figure 2.6: The tangent, normal, and binormal vectors define an orthogonal coordinate system along a space curve In Sects. 2.1 and 2.2 , we have introduced the tangent and normal vectors, which are orthogonal to each other and lie in the osculating plane.

## Calculus and Diﬀerential Geometry An Introduction

1.1 VECTOR CALCULUS (Derivative of Scalar & Vector. A Euclidean vector field V for which each vector V(p) is tangent to M at p is called a tangent vector field on M (Fig. 4.25). Frequently these vector fields are defined, not on all of M, but only on some region in M. As usual, we, Multivariate Calculus; Fall 2013 S. Jamshidi Deﬁnition 5.4.2 The directional derivative, denoted Dvf(x,y), is a derivative of a multivari-able function in the direction of a vector ~ v . It is the scalar projection of the gradient onto ~v . Dvf(x,y.

### Directional derivatives and gradient vectors (Sect.

Determining Curvature of a Curve Defined by a Vector. 2 CHAPTER 1. WHAT IS DIFFERENTIAL GEOMETRY? U f Figure 1.1: A chart Perhaps the user of such a map will be content to use the map to plot the shortest path between two points pand qin U. This path is called a geodesic., curve, a vector tangent to the curve and associated with the object must make a “full” rotation of 2 πradians or 360 . In other words, if we were to think of this tangent vector (of if you wish, a copy of it) as having its tail ﬁxed at the.

More generally, if v is any vector in Rm, then the product D pf(v) is called the directional derivative of f in the direction of v. This is something like a \partial derivative" in the direction of the vector v. The directional derivative D p(v) can be The function F always takes two arguments, the scalar independent variable, t, and the vector of dependent variables, y. A program that evaluates F(t;y) hould compute the derivatives of all the state variables and return them in

meaning? We say that a parametric curve and c: (a, b) —Y is footed at p max. flowline for V footed at p. VF V on Rn, maximal it extends other any A flowline for V footed at p is said to be V be a VF on Let Let flowline for V footed at 264 CHAPTER 9. CALCULUS ON EUCLIDEAN SPACE Deﬁnition 9.1. A tangent vector vp to Euclidean space Rn consists of a pair of elements v,p of Rn; v is called the vector part and p is called the point of application of vp. v v p

16 Vector Calculus 16.1 Vector Fields This chapter is concerned with applying calculus in the context of vector ﬁelds. A two-dimensional vector ﬁeld is a function f that maps each point (x,y) in R2 to a two-dimensional vector hu,vi, and NOTES ON DIFFERENTIAL GEOMETRY 3 the ﬁrst derivative of x: (6) t = dx/ds = x˙ Note that this is a unit vector precisely because we have assumed that the parameterization of the curve is unit-speed. The second derivative ¨x will

1.1 VECTOR CALCULUS (Derivative of Scalar & Vector Function) - Free download as Powerpoint Presentation (.ppt), PDF File (.pdf), Text File (.txt) or view presentation slides online. k m8 k m8 Search Search 2.3 Binormal vector and torsion Figure 2.6: The tangent, normal, and binormal vectors define an orthogonal coordinate system along a space curve In Sects. 2.1 and 2.2 , we have introduced the tangent and normal vectors, which are orthogonal to each other and lie in the osculating plane.

NOTES ON DIFFERENTIAL GEOMETRY 3 the ﬁrst derivative of x: (6) t = dx/ds = x˙ Note that this is a unit vector precisely because we have assumed that the parameterization of the curve is unit-speed. The second derivative ¨x will 2 Basic di erential geometry Conventions If UˆRm is open, V is a real (or complex) vector space (of nite dimension), and ’: U!V is a smooth function, then the partial derivative of ’with respect to x i is denoted in the following di erent ways,

Barrett O'Neill, in Elementary Differential Geometry (Second Edition), 2006 1.3 Directional Derivatives Associated with each tangent vector v p to R 3 is the straight line t → p + tv (see Example 4.2). If f is a differentiable function on R 3, then Multivariate Calculus; Fall 2013 S. Jamshidi Deﬁnition 5.4.2 The directional derivative, denoted Dvf(x,y), is a derivative of a multivari-able function in the direction of a vector ~ v . It is the scalar projection of the gradient onto ~v . Dvf(x,y

1.1. CURVES 2 lines), acceleration, and jerk that completely determine the path of the curve when we use some parameter t to travel along it. q v a j Tangent Line Most of the curves we study will be given as parametrized curves,i.e., Section 14.5, Directional derivatives and gradient vectors p. 331 (3/23/08) Estimating directional derivatives from level curves We could ﬁnd approximate values of directional derivatives from level curves by using the techniques of the

Chapter 4 Diﬀerentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f: D → R, where D is a subset of Rn, where n is the number of variables. These In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and

Sage Reference Manual: Differential Geometry of Curves and Surfaces, Release 8.9 (continued from previous page) sage: # Find minimum and max of the gauss curvature use the unit tangent vector ~tand a stepsize hto compute x 1 = x 0 + h~t and then apply Newton’s method to determine a point x 2 which satis es the implicit function, and has the same value of the p-coordinate as x 1. To do all

NOTES ON DIFFERENTIAL GEOMETRY 3 the ﬁrst derivative of x: (6) t = dx/ds = x˙ Note that this is a unit vector precisely because we have assumed that the parameterization of the curve is unit-speed. The second derivative ¨x will In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and

Engineering Mathematics - I Semester – 1 By Dr N V Nagendram UNIT – V Vector Differential Calculus Gradient, Divergence and Curl 1.01 Introduction 1.02 Vector Differentiation 1.03 Problems Exercise and solutions 1.04 1.1. CURVES 2 lines), acceleration, and jerk that completely determine the path of the curve when we use some parameter t to travel along it. q v a j Tangent Line Most of the curves we study will be given as parametrized curves,i.e.,

1.1. CURVES 2 lines), acceleration, and jerk that completely determine the path of the curve when we use some parameter t to travel along it. q v a j Tangent Line Most of the curves we study will be given as parametrized curves,i.e., 264 CHAPTER 9. CALCULUS ON EUCLIDEAN SPACE Deﬁnition 9.1. A tangent vector vp to Euclidean space Rn consists of a pair of elements v,p of Rn; v is called the vector part and p is called the point of application of vp. v v p

Concrete example of the derivative of a vector valued function to better understand what it means If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter 2013/09/23 · Compute unit normal vector, unit tangent vector, and curvature. Description of basic geometry and an example. Compute unit normal vector, unit tangent vector, and curvature. Description of basic geometry and an example. Sign in

1.1. CURVES 2 lines), acceleration, and jerk that completely determine the path of the curve when we use some parameter t to travel along it. q v a j Tangent Line Most of the curves we study will be given as parametrized curves,i.e., Engineering Mathematics - I Semester – 1 By Dr N V Nagendram UNIT – V Vector Differential Calculus Gradient, Divergence and Curl 1.01 Introduction 1.02 Vector Differentiation 1.03 Problems Exercise and solutions 1.04

• “Discrete Differential‐Geometry Operators for Triangulated 2‐ Manifolds”, Meyer et al., ’02 • “Restricted Delaunay triangulations and normal cycle”, Cohen‐Steiner et al., SoCG ‘03 • “On the convergence of metric and geometric Multivariate Calculus; Fall 2013 S. Jamshidi Deﬁnition 5.4.2 The directional derivative, denoted Dvf(x,y), is a derivative of a multivari-able function in the direction of a vector ~ v . It is the scalar projection of the gradient onto ~v . Dvf(x,y

meaning? We say that a parametric curve and c: (a, b) —Y is footed at p max. flowline for V footed at p. VF V on Rn, maximal it extends other any A flowline for V footed at p is said to be V be a VF on Let Let flowline for V footed at 2.3 Binormal vector and torsion Figure 2.6: The tangent, normal, and binormal vectors define an orthogonal coordinate system along a space curve In Sects. 2.1 and 2.2 , we have introduced the tangent and normal vectors, which are orthogonal to each other and lie in the osculating plane.

### Calculus on Euclidean Space unito.it

13.2 Calculus with vector functions. 16 Vector Calculus 16.1 Vector Fields This chapter is concerned with applying calculus in the context of vector ﬁelds. A two-dimensional vector ﬁeld is a function f that maps each point (x,y) in R2 to a two-dimensional vector hu,vi, and, Sage Reference Manual: Differential Geometry of Curves and Surfaces, Release 8.9 (continued from previous page) sage: # Find minimum and max of the gauss curvature.

### 2.1 Arc length and tangent vector

Directional Derivatives and The Gradient Vector. Engineering Mathematics - I Semester – 1 By Dr N V Nagendram UNIT – V Vector Differential Calculus Gradient, Divergence and Curl 1.01 Introduction 1.02 Vector Differentiation 1.03 Problems Exercise and solutions 1.04 In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and.

In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can In vector (or multivariable) calculus, we will deal with functions of two or three variables (usually x,y or x,y,z, respectively). The graph of a function of two variables, say, z=f(x,y), lies in Euclidean space, which in the Cartesian a,b,c).

Introduction to Differential Geometry Robert Bartnik January 1995 These notes are designed to give a heuristic guide to many of the basic constructions of differential geometry. They are by no means complete; nor are they at all 2018/05/31 · Section 6-7 : Calculus with Vector Functions In this section we need to talk briefly about limits, derivatives and integrals of vector functions. As you will see, these behave in a fairly predictable manner. We will be doing all of the work

More generally, if v is any vector in Rm, then the product D pf(v) is called the directional derivative of f in the direction of v. This is something like a \partial derivative" in the direction of the vector v. The directional derivative D p(v) can be 2 2.2. Vector ﬁelds. In Euclidean (and Riemannian) geometry, a vector at a point has direction and magnitude. In differential geometry, there are two kinds of vectors and each of these only has some of the familiar properties of vectors

2018/05/31 · Section 6-7 : Calculus with Vector Functions In this section we need to talk briefly about limits, derivatives and integrals of vector functions. As you will see, these behave in a fairly predictable manner. We will be doing all of the work curve, a vector tangent to the curve and associated with the object must make a “full” rotation of 2 πradians or 360 . In other words, if we were to think of this tangent vector (of if you wish, a copy of it) as having its tail ﬁxed at the

In vector (or multivariable) calculus, we will deal with functions of two or three variables (usually x,y or x,y,z, respectively). The graph of a function of two variables, say, z=f(x,y), lies in Euclidean space, which in the Cartesian a,b,c). Vector ﬁelds: motivation, deﬁnition Vector ﬁelds: why, where? A vector ﬁeld arises in a situation where, for some reason, there is a direction and a magnitude assigned to each point of the space or of a surface, typically examples are

A Euclidean vector field V for which each vector V(p) is tangent to M at p is called a tangent vector field on M (Fig. 4.25). Frequently these vector fields are defined, not on all of M, but only on some region in M. As usual, we Multivariate Calculus; Fall 2013 S. Jamshidi Deﬁnition 5.4.2 The directional derivative, denoted Dvf(x,y), is a derivative of a multivari-able function in the direction of a vector ~ v . It is the scalar projection of the gradient onto ~v . Dvf(x,y

In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can Concrete example of the derivative of a vector valued function to better understand what it means If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter

Section 14.5, Directional derivatives and gradient vectors p. 331 (3/23/08) Estimating directional derivatives from level curves We could ﬁnd approximate values of directional derivatives from level curves by using the techniques of the 2 2.2. Vector ﬁelds. In Euclidean (and Riemannian) geometry, a vector at a point has direction and magnitude. In differential geometry, there are two kinds of vectors and each of these only has some of the familiar properties of vectors

A Euclidean vector field V for which each vector V(p) is tangent to M at p is called a tangent vector field on M (Fig. 4.25). Frequently these vector fields are defined, not on all of M, but only on some region in M. As usual, we Home Calculators Calculus III Calculators Math Problem Solver (all calculators) Unit Tangent Vector Calculator The calculator will find the unit tangent vector of a vector-valued function at the given point, with steps shown.

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The voltage created in inclined thin anisotropic films illuminated by laser light is significantly affected by a photovoltaic component.

A significant transverse voltage (more than $10\phantom{\rule{0.333em}{0ex}}\text{V}$) has been observed for several inclined-oriented anisotropic thin films when illuminated by laser light. This effect has been attributed to the off-diagonal thermoelectric effect, in which a temperature gradient orthogonal to the film plane induces a voltage parallel to the plane. However, recent results that deviate from the expected behavior of the thermoelectric effect have caused researchers to propose that the induced voltage may be due to an optical and hence nonthermal effect.

Writing in Physical Review B, Kouhei Takahashi and co-workers at Panasonic Corporation in Kyoto, Japan, present measurements of laser-induced voltages in ${\text{Ca}}_{x}{\text{Co0}}_{2}$ ($x\sim 0.5$) thin films at different laser wavelengths. The signals behave as expected for energies lower than the band gap of the film, but for energies above the band gap, the signal is smaller than what would be expected from the off-diagonal thermoelectric effect alone. This deviation is consistent with an interband transition, indicating that a small photovoltaic effect is suppressing the voltage signal. By revealing the nature of the different effects on the signal, this experiment helps resolve the controversy of the responsible effect. The results have bearing on applications such as sensitive thermal sensors and thermoelectric generators. – Brad Rubin

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# Stuck with an integration

How do you integrate $$\int_0^1 y^{k_1} (1 - (1-y)^{\alpha})^{k_2}\,\mathrm{d}y$$ where $k_1$ and $k_2$ are non-negative integers and $\alpha>1$?

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Expand with binomial theorem and use beta functions. –  anon Jan 27 '12 at 6:57
@anon $k1$ and $k2$ arent necessarily positive integers, no? –  Ravi Donepudi Jan 27 '12 at 7:06
actually k1 and k2 are positive integers(including 0).. Also, does the integral become more tractable if alpha is restricted to rational numbers –  Nischal Jan 27 '12 at 7:16

$$I=\int\limits_0^1 {{y^a}{{\left( {1 - {{\left( {1 - y} \right)}^b }} \right)}^c}} dy$$

$$1-y=u$$ $$I=\int\limits_0^1 {{{\left( {1 - u} \right)}^a}{{\left( {1 - {u^b}} \right)}^c}} du$$

$${\left( {1 - {u^b }} \right)^c} = \sum\limits_{k = 0}^c {{{\left( { - 1} \right)}^{bk}}{c \choose k}{u^{bk}}}$$

$$I=\sum\limits_{k = 0}^c {{{\left( { - 1} \right)}^{bk}}{c \choose k}\int\limits_0^1 {{{\left( {1 - u} \right)}^a}{u^{bk}}} du}$$

$$\int\limits_0^1 {{{\left( {1 - u} \right)}^a}{u^{bk}}} du = B\left( {a+1,bk+1} \right)$$

$$\sum\limits_{k = 0}^c {{{\left( { - 1} \right)}^{bk}}{c \choose k}B\left( {a+1,bk+1} \right)}$$

Maybe there is some multiple argument formula you can use to simplify things.

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Precalculus : Convert Polar Equations To Rectangular Form

Example Questions

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Example Question #21 : Convert Polar Equations To Rectangular Form

Write the equation for  in rectangular form

Explanation:

First multiply both sides by cosine

Now we can make the substitutions and

square both sides

multiply both sides by the denominator

distribute on the left side

subtract from both sides

factor out y squared

divide both sides by

take the square root

Example Question #21 : Convert Polar Equations To Rectangular Form

Which is equivalent to in rectangular form?

Explanation:

Multiply both sides by r squared

Now we can substitute and

square both sides

Example Question #23 : Convert Polar Equations To Rectangular Form

Which is equivalent to in rectangular form?

Explanation:

First multiply both sides by r:

Now make the substitutions and :

subtract x squared and y squared from both sides

square both sides

Example Question #24 : Convert Polar Equations To Rectangular Form

Write in rectangular form

Explanation:

multiply both sides by r:

Now we can make the substitutions , , and :

If we subtract the terms on the right from both sides, we can complete the square twice and make this into a regular circle equation:

To complete the square for x, add 1, since .

To complete the square for y, add since .

re-write the left side as binomials squared:

Example Question #25 : Convert Polar Equations To Rectangular Form

Write the equation for in rectangular form.

Explanation:

Multiply both sides by r:

Now we can substitute in , , and .

if we subtract both terms on the right from both sides, we can complete the square twice to get this into the normal form for a circle.

To complete the square for x, add 1 to both sides, since

To complete the square for y, add 16 to both sides, since :

Re-write the left side as two binomials squared

Example Question #26 : Convert Polar Equations To Rectangular Form

Write the equation for in rectangular form

Explanation:

To convert, make the substitutions and :

square both sides

multiply both sides by

subtract  from both sides

subtract from both sides

factor our

divide both sides  by

take the square root of both sides

Example Question #27 : Convert Polar Equations To Rectangular Form

Convert the equation to rectangular form

Explanation:

Multiply both sides by :

distribute

multiply both sides by r

convert to rectangular by making the substitutions , , and :

subtract 8x from both sides and combine like terms

complete the square for x by adding 16 to both sides

divide both sides by 16

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# Quick start using graph-tool¶

The graph_tool module provides a Graph class and several algorithms that operate on it. The internals of this class, and of most algorithms, are written in C++ for performance, using the Boost Graph Library.

The module must be of course imported before it can be used. The package is subdivided into several sub-modules. To import everything from all of them, one can do:

>>> from graph_tool.all import *

In the following, it will always be assumed that the previous line was run.

## Creating and manipulating graphs¶

An empty graph can be created by instantiating a Graph class:

>>> g = Graph()

By default, newly created graphs are always directed. To construct undirected graphs, one must pass a value to the directed parameter:

>>> ug = Graph(directed=False)

A graph can always be switched on-the-fly from directed to undirected (and vice versa), with the set_directed() method. The “directedness” of the graph can be queried with the is_directed() method,

>>> ug = Graph()
>>> ug.set_directed(False)
>>> assert ug.is_directed() == False

A graph can also be created by providing another graph, in which case the entire graph (and its internal property maps, see Property maps) is copied:

>>> g1 = Graph()
>>> # ... construct g1 ...
>>> g2 = Graph(g1)                 # g1 and g2 are copies

Above, g2 is a “deep” copy of g1, i.e. any modification of g2 will not affect g1.

Once a graph is created, it can be populated with vertices and edges. A vertex can be added with the add_vertex() method, which returns an instance of a Vertex class, also called a vertex descriptor. For instance, the following code creates two vertices, and returns vertex descriptors stored in the variables v1 and v2.

>>> v1 = g.add_vertex()

Edges can be added in an analogous manner, by calling the add_edge() method, which returns an edge descriptor (an instance of the Edge class):

>>> e = g.add_edge(v1, v2)

The above code creates a directed edge from v1 to v2. We can visualize the graph we created so far with the graph_draw() function.

>>> graph_draw(g, vertex_text=g.vertex_index, vertex_font_size=18,
...            output_size=(200, 200), output="two-nodes.png")
<...>

A simple directed graph with two vertices and one edge, created by the commands above.

With vertex and edge descriptors, one can examine and manipulate the graph in an arbitrary manner. For instance, in order to obtain the out-degree of a vertex, we can simply call the out_degree() method:

>>> print(v1.out_degree())
1

Analogously, we could have used the in_degree() method to query the in-degree.

Note

For undirected graphs, the “out-degree” is synonym for degree, and in this case the in-degree of a vertex is always zero.

Edge descriptors have two useful methods, source() and target(), which return the source and target vertex of an edge, respectively.

>>> print(e.source(), e.target())
0 1

The add_vertex() method also accepts an optional parameter which specifies the number of vertices to create. If this value is greater than 1, it returns an iterator on the added vertex descriptors:

>>> vlist = g.add_vertex(10)
>>> print(len(list(vlist)))
10

Each vertex in a graph has an unique index, which is always between $$0$$ and $$N-1$$, where $$N$$ is the number of vertices. This index can be obtained by using the vertex_index attribute of the graph (which is a property map, see Property maps), or by converting the vertex descriptor to an int.

>>> v = g.add_vertex()
>>> print(g.vertex_index[v])
12
>>> print(int(v))
12

Edges and vertices can also be removed at any time with the remove_vertex() and remove_edge() methods,

>>> g.remove_edge(e)                               # e no longer exists
>>> g.remove_vertex(v2)                # the second vertex is also gone

Note

Removing a vertex is typically an $$O(N)$$ operation. The vertices are internally stored in a STL vector, so removing an element somewhere in the middle of the list requires the shifting of the rest of the list. Thus, fast $$O(1)$$ removals are only possible either if one can guarantee that only vertices in the end of the list are removed (the ones last added to the graph), or if the relative vertex ordering is invalidated. The latter behavior can be achieved by passing the option fast == True, to remove_vertex(), which causes the vertex being deleted to be ‘swapped’ with the last vertex (i.e. with the largest index), which will in turn inherit the index of the vertex being deleted.

Warning

Because of the above, removing a vertex with an index smaller than $$N-1$$ will invalidate either the last (fast = True) or all (fast = False) descriptors pointing to vertices with higher index.

As a consequence, if more than one vertex is to be removed at a given time, they should always be removed in decreasing index order:

# 'del_list' is a list of vertex descriptors
for v in reversed(sorted(del_list)):
g.remove_vertex(v)

Alternatively (and preferably), a list (or any iterable) may be passed directly as the vertex parameter of the remove_vertex() function, and the above is performed internally (in C++).

Note that property map values (see Property maps) are unaffected by the index changes due to vertex removal, as they are modified accordingly by the library.

Note

Removing an edge is an $$O(k_{s} + k_{t})$$ operation, where $$k_{s}$$ is the out-degree of the source vertex, and $$k_{t}$$ is the in-degree of the target vertex. This can be made faster by setting set_fast_edge_removal() to True, in which case it becomes $$O(1)$$, at the expense of additional data of size $$O(E)$$.

No edge descriptors are ever invalidated after edge removal, with the exception of the edge being removed.

Since vertices are uniquely identifiable by their indexes, there is no need to keep the vertex descriptor lying around to access them at a later point. If we know its index, we can obtain the descriptor of a vertex with a given index using the vertex() method,

>>> v = g.vertex(8)

which takes an index, and returns a vertex descriptor. Edges cannot be directly obtained by its index, but if the source and target vertices of a given edge are known, it can be retrieved with the edge() method

>>> g.add_edge(g.vertex(2), g.vertex(3))
<...>
>>> e = g.edge(2, 3)

Another way to obtain edge or vertex descriptors is to iterate through them, as described in section Iterating over vertices and edges. This is in fact the most useful way of obtaining vertex and edge descriptors.

Like vertices, edges also have unique indexes, which are given by the edge_index property:

>>> e = g.add_edge(g.vertex(0), g.vertex(1))
>>> print(g.edge_index[e])
1

Differently from vertices, edge indexes do not necessarily conform to any specific range. If no edges are ever removed, the indexes will be in the range $$[0, E-1]$$, where $$E$$ is the number of edges, and edges added earlier have lower indexes. However if an edge is removed, its index will be “vacant”, and the remaining indexes will be left unmodified, and thus will not all lie in the range $$[0, E-1]$$. If a new edge is added, it will reuse old indexes, in an increasing order.

### Iterating over vertices and edges¶

Algorithms must often iterate through vertices, edges, out-edges of a vertex, etc. The Graph and Vertex classes provide different types of iterators for doing so. The iterators always point to edge or vertex descriptors.

#### Iterating over all vertices or edges¶

In order to iterate through all the vertices or edges of a graph, the vertices() and edges() methods should be used:

for v in g.vertices():
print(v)
for e in g.edges():
print(e)

The code above will print the vertices and edges of the graph in the order they are found.

#### Iterating over the neighborhood of a vertex¶

The out- and in-edges of a vertex, as well as the out- and in-neighbors can be iterated through with the out_edges(), in_edges(), out_neighbors() and in_neighbors() methods, respectively.

from itertools import izip
for v in g.vertices():
for e in v.out_edges():
print(e)
for w in v.out_neighbors():
print(w)

# the edge and neighbors order always match
for e, w in izip(v.out_edges(), v.out_neighbors()):
assert e.target() == w

The code above will print the out-edges and out-neighbors of all vertices in the graph.

Warning

You should never remove vertex or edge descriptors when iterating over them, since this invalidates the iterators. If you plan to remove vertices or edges during iteration, you must first store them somewhere (such as in a list) and remove them only after no iterator is being used. Removal during iteration will cause bad things to happen.

#### Fast iteration over vertices and edges¶

While convenient, looping over the graph as described in the previous section is not the most efficient approach. This is because the loops are performed in pure Python, and hence it undermines the main feature of the library, which is the offloading of loops from Python to C++. Following the numpy philosophy, graph_tool also provides an array-based interface that avoids loops in Python. This is done with the get_vertices(), get_edges(), get_out_edges(), get_in_edges(), get_out_neighbors(), get_in_neighbors(), get_out_degrees() and get_in_degrees() methods, which return numpy.ndarray instances instead of iterators.

For example, using this interface we can get the out-degree of each node via:

print(g.get_out_degrees(g.get_vertices()))

[1 0 1 0 0 0 0 0 0 0 0 0]

or the sum of the product of the in and out-degrees of the endpoints of each edge with:

edges = g.get_edges()
print((edges[:,0] * edges[:,1]).sum())

6

## Property maps¶

Property maps are a way of associating additional information to the vertices, edges or to the graph itself. There are thus three types of property maps: vertex, edge and graph. All of them are handled by the same class, PropertyMap. Each created property map has an associated value type, which must be chosen from the predefined set:

Type name Alias
bool uint8_t
int16_t short
int32_t int
int64_t long, long long
double float
long double
string
vector<bool> vector<uint8_t>
vector<int16_t> vector<short>
vector<int32_t> vector<int>
vector<int64_t> vector<long>, vector<long long>
vector<double> vector<float>
vector<long double>
vector<string>
python::object object

New property maps can be created for a given graph by calling the new_vertex_property(), new_edge_property(), or new_graph_property(), for each map type. The values are then accessed by vertex or edge descriptors, or the graph itself, as such:

from itertools import izip
from numpy.random import randint

g = Graph()
for s,t in izip(randint(0, 100, 100), randint(0, 100, 100)):

vprop_double = g.new_vertex_property("double")            # Double-precision floating point
vprop_double[g.vertex(10)] = 3.1416

vprop_vint = g.new_vertex_property("vector<int>")         # Vector of ints
vprop_vint[g.vertex(40)] = [1, 3, 42, 54]

eprop_dict = g.new_edge_property("object")                # Arbitrary python object.
eprop_dict[g.edges().next()] = {"foo": "bar", "gnu": 42}  # In this case, a dict.

gprop_bool = g.new_graph_property("bool")                  # Boolean
gprop_bool[g] = True

Property maps with scalar value types can also be accessed as a numpy.ndarray, with the get_array() method, or the a attribute, i.e.,

from numpy.random import random

# this assigns random values to the vertex properties
vprop_double.get_array()[:] = random(g.num_vertices())

# or more conveniently (this is equivalent to the above)
vprop_double.a = random(g.num_vertices())

### Internal property maps¶

Any created property map can be made “internal” to the corresponding graph. This means that it will be copied and saved to a file together with the graph. Properties are internalized by including them in the graph’s dictionary-like attributes vertex_properties, edge_properties or graph_properties (or their aliases, vp, ep or gp, respectively). When inserted in the graph, the property maps must have an unique name (between those of the same type):

>>> eprop = g.new_edge_property("string")
>>> g.edge_properties["some name"] = eprop
>>> g.list_properties()
some name      (edge)    (type: string)

Internal graph property maps behave slightly differently. Instead of returning the property map object, the value itself is returned from the dictionaries:

>>> gprop = g.new_graph_property("int")
>>> g.graph_properties["foo"] = gprop   # this sets the actual property map
>>> g.graph_properties["foo"] = 42      # this sets its value
>>> print(g.graph_properties["foo"])
42
>>> del g.graph_properties["foo"]       # the property map entry is deleted from the dictionary

For convenience, the internal property maps can also be accessed via attributes:

>>> vprop = g.new_vertex_property("double")
>>> g.vp.foo = vprop                        # equivalent to g.vertex_properties["foo"] = vprop
>>> v = g.vertex(0)
>>> g.vp.foo[v] = 3.14
>>> print(g.vp.foo[v])
3.14

## Graph I/O¶

Graphs can be saved and loaded in four formats: graphml, dot, gml and a custom binary format gt (see The gt file format).

Warning

The binary format gt and the text-based graphml are the preferred formats, since they are by far the most complete. Both these formats are equally complete, but the gt format is faster and requires less storage.

The dot and gml formats are fully supported, but since they contain no precise type information, all properties are read as strings (or also as double, in the case of gml), and must be converted by hand to the desired type. Therefore you should always use either gt or graphml, since they implement an exact bit-for-bit representation of all supported Property maps types, except when interfacing with other software, or existing data, which uses dot or gml.

A graph can be saved or loaded to a file with the save and load methods, which take either a file name or a file-like object. A graph can also be loaded from disc with the load_graph() function, as such:

g = Graph()
#  ... fill the graph ...
g.save("my_graph.xml.gz")
# g and g2 should be copies of each other

Graph classes can also be pickled with the pickle module.

## An Example: Building a Price Network¶

A Price network is the first known model of a “scale-free” graph, invented in 1976 by de Solla Price. It is defined dynamically, where at each time step a new vertex is added to the graph, and connected to an old vertex, with probability proportional to its in-degree. The following program implements this construction using graph-tool.

Note

Note that it would be much faster just to use the price_network() function, which is implemented in C++, as opposed to the script below which is in pure python. The code below is merely a demonstration on how to use the library.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 #! /usr/bin/env python # We will need some things from several places from __future__ import division, absolute_import, print_function import sys if sys.version_info < (3,): range = xrange import os from pylab import * # for plotting from numpy.random import * # for random sampling seed(42) # We need to import the graph_tool module itself from graph_tool.all import * # let's construct a Price network (the one that existed before Barabasi). It is # a directed network, with preferential attachment. The algorithm below is # very naive, and a bit slow, but quite simple. # We start with an empty, directed graph g = Graph() # We want also to keep the age information for each vertex and edge. For that # let's create some property maps v_age = g.new_vertex_property("int") e_age = g.new_edge_property("int") # The final size of the network N = 100000 # We have to start with one vertex v = g.add_vertex() v_age[v] = 0 # we will keep a list of the vertices. The number of times a vertex is in this # list will give the probability of it being selected. vlist = [v] # let's now add the new edges and vertices for i in range(1, N): # create our new vertex v = g.add_vertex() v_age[v] = i # we need to sample a new vertex to be the target, based on its in-degree + # 1. For that, we simply randomly sample it from vlist. i = randint(0, len(vlist)) target = vlist[i] # add edge e = g.add_edge(v, target) e_age[e] = i # put v and target in the list vlist.append(target) vlist.append(v) # now we have a graph! # let's do a random walk on the graph and print the age of the vertices we find, # just for fun. v = g.vertex(randint(0, g.num_vertices())) while True: print("vertex:", int(v), "in-degree:", v.in_degree(), "out-degree:", v.out_degree(), "age:", v_age[v]) if v.out_degree() == 0: print("Nowhere else to go... We found the main hub!") break n_list = [] for w in v.out_neighbors(): n_list.append(w) v = n_list[randint(0, len(n_list))] # let's save our graph for posterity. We want to save the age properties as # well... To do this, they must become "internal" properties: g.vertex_properties["age"] = v_age g.edge_properties["age"] = e_age # now we can save it g.save("price.xml.gz") # Let's plot its in-degree distribution in_hist = vertex_hist(g, "in") y = in_hist[0] err = sqrt(in_hist[0]) err[err >= y] = y[err >= y] - 1e-2 figure(figsize=(6,4)) errorbar(in_hist[1][:-1], in_hist[0], fmt="o", yerr=err, label="in") gca().set_yscale("log") gca().set_xscale("log") gca().set_ylim(1e-1, 1e5) gca().set_xlim(0.8, 1e3) subplots_adjust(left=0.2, bottom=0.2) xlabel("$k_{in}$") ylabel("$NP(k_{in})$") tight_layout() savefig("price-deg-dist.pdf") savefig("price-deg-dist.svg")

The following is what should happen when the program is run.

vertex: 36063 in-degree: 0 out-degree: 1 age: 36063
vertex: 9075 in-degree: 4 out-degree: 1 age: 9075
vertex: 5967 in-degree: 3 out-degree: 1 age: 5967
vertex: 1113 in-degree: 7 out-degree: 1 age: 1113
vertex: 25 in-degree: 84 out-degree: 1 age: 25
vertex: 10 in-degree: 541 out-degree: 1 age: 10
vertex: 5 in-degree: 140 out-degree: 1 age: 5
vertex: 2 in-degree: 459 out-degree: 1 age: 2
vertex: 1 in-degree: 520 out-degree: 1 age: 1
vertex: 0 in-degree: 210 out-degree: 0 age: 0
Nowhere else to go... We found the main hub!

Below is the degree distribution, with $$10^5$$ nodes (in order to the asymptotic behavior to be even clearer, the number of vertices needs to be increased to something like $$10^6$$ or $$10^7$$).

In-degree distribution of a price network with $$10^5$$ nodes.

We can draw the graph to see some other features of its topology. For that we use the graph_draw() function.

g = load_graph("price.xml.gz")
age = g.vertex_properties["age"]

pos = sfdp_layout(g)
graph_draw(g, pos, output_size=(1000, 1000), vertex_color=[1,1,1,0],
vertex_fill_color=age, vertex_size=1, edge_pen_width=1.2,
vcmap=matplotlib.cm.gist_heat_r, output="price.png")

A Price network with $$10^5$$ nodes. The vertex colors represent the age of the vertex, from older (red) to newer (black).

## Graph filtering¶

One of the very nice features of graph-tool is the “on-the-fly” filtering of edges and/or vertices. Filtering means the temporary masking of vertices/edges, which are in fact not really removed, and can be easily recovered. Vertices or edges which are to be filtered should be marked with a PropertyMap with value type bool, and then set with set_vertex_filter() or set_edge_filter() methods. By default, vertex or edges with value “1” are kept in the graphs, and those with value “0” are filtered out. This behaviour can be modified with the inverted parameter of the respective functions. All manipulation functions and algorithms will work as if the marked edges or vertices were removed from the graph, with minimum overhead.

Note

It is important to emphasize that the filtering functionality does not add any overhead when the graph is not being filtered. In this case, the algorithms run just as fast as if the filtering functionality didn’t exist.

Here is an example which obtains the minimum spanning tree of a graph, using the function min_spanning_tree() and edge filtering.

g, pos = triangulation(random((500, 2)) * 4, type="delaunay")
tree = min_spanning_tree(g)
graph_draw(g, pos=pos, edge_color=tree, output="min_tree.svg")

The tree property map has a bool type, with value “1” if the edge belongs to the tree, and “0” otherwise. Below is an image of the original graph, with the marked edges.

We can now filter out the edges which don’t belong to the minimum spanning tree.

g.set_edge_filter(tree)
graph_draw(g, pos=pos, output="min_tree_filtered.svg")

This is how the graph looks when filtered:

Everything should work transparently on the filtered graph, simply as if the masked edges were removed. For instance, the following code will calculate the betweenness() centrality of the edges and vertices, and draws them as colors and line thickness in the graph.

bv, be = betweenness(g)
be.a /= be.a.max() / 5
graph_draw(g, pos=pos, vertex_fill_color=bv, edge_pen_width=be,
output="filtered-bt.svg")

The original graph can be recovered by setting the edge filter to None.

g.set_edge_filter(None)
bv, be = betweenness(g)
be.a /= be.a.max() / 5
graph_draw(g, pos=pos, vertex_fill_color=bv, edge_pen_width=be,
output="nonfiltered-bt.svg")

Everything works in analogous fashion with vertex filtering.

Additionally, the graph can also have its edges reversed with the set_reversed() method. This is also an $$O(1)$$ operation, which does not really modify the graph.

As mentioned previously, the directedness of the graph can also be changed “on-the-fly” with the set_directed() method.

### Graph views¶

It is often desired to work with filtered and unfiltered graphs simultaneously, or to temporarily create a filtered version of graph for some specific task. For these purposes, graph-tool provides a GraphView class, which represents a filtered “view” of a graph, and behaves as an independent graph object, which shares the underlying data with the original graph. Graph views are constructed by instantiating a GraphView class, and passing a graph object which is supposed to be filtered, together with the desired filter parameters. For example, to create a directed view of the graph g constructed above, one should do:

>>> ug = GraphView(g, directed=True)
>>> ug.is_directed()
True

Graph views also provide a much more direct and convenient approach to vertex/edge filtering: To construct a filtered minimum spanning tree like in the example above, one must only pass the filter property as the “efilt” parameter:

>>> tv = GraphView(g, efilt=tree)

Note that this is an $$O(1)$$ operation, since it is equivalent (in speed) to setting the filter in graph g directly, but in this case the object g remains unmodified.

Like above, the result should be the isolated minimum spanning tree:

>>> bv, be = betweenness(tv)
>>> be.a /= be.a.max() / 5
>>> graph_draw(tv, pos=pos, vertex_fill_color=bv,
...            edge_pen_width=be, output="mst-view.svg")
<...>

A view of the Delaunay graph, isolating only the minimum spanning tree.

Note

GraphView objects behave exactly like regular Graph objects. In fact, GraphView is a subclass of Graph. The only difference is that a GraphView object shares its internal data with its parent Graph class. Therefore, if the original Graph object is modified, this modification will be reflected immediately in the GraphView object, and vice versa.

For even more convenience, one can supply a function as filter parameter, which takes a vertex or an edge as single parameter, and returns True if the vertex/edge should be kept and False otherwise. For instance, if we want to keep only the most “central” edges, we can do:

>>> bv, be = betweenness(g)
>>> u = GraphView(g, efilt=lambda e: be[e] > be.a.max() / 2)

This creates a graph view u which contains only the edges of g which have a normalized betweenness centrality larger than half of the maximum value. Note that, differently from the case above, this is an $$O(E)$$ operation, where $$E$$ is the number of edges, since the supplied function must be called $$E$$ times to construct a filter property map. Thus, supplying a constructed filter map is always faster, but supplying a function can be more convenient.

The graph view constructed above can be visualized as

>>> be.a /= be.a.max() / 5
>>> graph_draw(u, pos=pos, vertex_fill_color=bv, output="central-edges-view.svg")
<...>

A view of the Delaunay graph, isolating only the edges with normalized betweenness centrality larger than 0.01.

#### Composing graph views¶

Since graph views are regular graphs, one can just as easily create graph views of graph views. This provides a convenient way of composing filters. For instance, in order to isolate the minimum spanning tree of all vertices of the example above which have a degree larger than four, one can do:

>>> u = GraphView(g, vfilt=lambda v: v.out_degree() > 4)
>>> tree = min_spanning_tree(u)
>>> u = GraphView(u, efilt=tree)

The resulting graph view can be visualized as

>>> graph_draw(u, pos=pos, output="composed-filter.svg")
<...>

A composed view, obtained as the minimum spanning tree of all vertices in the graph which have a degree larger than four.


text:
Menpo Playground
an isolated, standalone build of Menpo.

A Menpo Playground is an archived directory you can download that contains:

1. A full isolated Python installation containing the Menpo Project with all its dependencies ready to go
2. A set of notebooks that you can run immediately to see how to use Menpo
3. Two command line tools, menpofit, and menpodetect (provided by menpocli), that can be used to locate bounding boxes and landmarks in challenging in-the-wild facial images
4. Detailed documentation on how to use the provided tools

There is no installation procedure for a playground, just download, extract, and use. When you are done with it, just delete the folder.

A playground is great from newcomers to Menpo, or just for anyone needing a hassle-free standalone build of Menpo (say, to deploy on a server).


text:
# Is Kullback-Leibler divergence a general measure of the difference of two probability distributions? Relation to Kolmogorov-Smirnov Test?

I am reading 'Deep Learning' by Goodfellow et al and in its review of Information Theory states that the Kullback-Leibler divergence can measure how different two distributions are, without any further qualification.

However, to me it seems that KL divergence measures the difference of two distributions from the point of view of the Information Value and not in general.

If I understand well, two symmetric distributions around their mean should have the same Information Value -- the amount of 'certainty' and 'surprise' from messages relating to Events generated from these two distributions should be the same given their symmetry. In the same time these two symmetric distributions would be very different, one representing a high mass of probability for low values and the other for high values.

Is my understanding correct?

And finally how Kullback_leibler divergence compares to the Kolmogorov-Smirnov Test?

• There's a bunch of differences - for instance, the K-S statistic is only defined for distributions on $\mathbb{R},$ whilst the K-L is defined for arbitrary alphabets. Distributional discrepancy (which is morally K-S but without to $\mathbb{R}$) can be upper bounded by total variation distance, at which point you can invoke Pinsker's inequality to relate it to K-L. As always, which divergence is appropriate/useful for a particular problem will vary (for instance, discrepancy induces a rather strong topology). This might be a useful read. – stochasticboy321 Jun 3 '18 at 7:41


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Mean absolute deviation calculator for ungrouped data

Mean absolute deviation for ungrouped data Mean absolute deviation is another measure of dispersion. MAD is an absolute measure of dispersion. Let $x_i, i=1,2, \cdots , n$ be $n$ observations. The mean of $X$ is denoted by $\overline{x}$ and is given by $$\begin{eqnarray*} \overline{x}& =&\frac{1}{n}\sum_{i=1}^{n}x_i \end{eqnarray*}$$ The mean absolute deviation about mean is …

Percentiles Calculator for grouped data with examples

Percentiles for grouped data Percentiles are the values which divide whole distribution into hundred equal parts. They are 99 in numbers namely $P_1, P_2, \cdots, P_{99}$. Here $P_1$ is first percentile, $P_2$ is second percentile and so on. For discrete frequency distribution, the formula for $i^{th}$ percentile is $P_i =\bigg(\dfrac{i(N)}{100}\bigg)^{th}$ value, $i=1,2,\cdots, 99$ where, $N$ …


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# Compute the probability of only one candidate arriving in 2-hour interval of time.

Question description

candidates arrive at ABC company for job interview at the rate of 3 per hour.

The number of candidates arriving follows a Poisson distribution:

Where,

𝑓(𝑥) = the probability of 𝑥 candidates arriving in an interval

𝜇 = mean number of candidates arriving in an interval

a. Compute the probability of exactly 5 candidates arriving in 6-hour interval of time

b. Compute the probability of at most 2 candidates arriving in 6-hour interval of time

c. Compute the probability of only one candidate arriving in 2-hour interval of time.


text:
# zbMATH — the first resource for mathematics

Adiabatic limits and the spectrum of the Laplacian on foliated manifolds. (English) Zbl 1159.58018
Burghelea, Dan (ed.) et al., $$C^*$$-algebras and elliptic theory II. Selected papers of the international conference, Bȩdlewo, Poland, January 2006. Basel: Birkhäuser (ISBN 978-3-7643-8603-0/hbk). Trends in Mathematics, 123-144 (2008).
Let $$\mathcal{F}$$ be a smooth foliation on a manifold $$M$$ with a Riemannian metric $$g$$. The tangent bundle decomposes as direct sum of the subbundles of vectors tangent and perpendicular to the leaves, respectively: $$TM=F\oplus H$$. This yields a corresponding decomposition of the metric tensor, $$g=g_F+g_H$$, and we can consider the family of metrics $$g_h=g_F+h^{-2}g_H$$ for $$h>0$$. The study of the asymptotic behaviour of $$(M,g_h)$$ as $$h\to0$$, called adiabatic limit, was begun by E. Witten [Commun. Math. Phys. 100, 197–229 (1985; Zbl 0581.58038)], and continued by many authors.
In the present paper, the authors consider the adiabatic limit of the spectrum of the Laplacian as follows. Let $$\Delta_h$$ denote the laplacian defined by $$g_h$$, and let $0\leq\lambda_0^r(h)\leq\lambda_1^r(h)\leq\lambda_2^r(h)\leq\cdots$ be its spectrum on $$r$$-forms (with multiplicities); each function $$\lambda_i^r(h)$$ is called a spectral branch. The main results about the asymptotics of the spectral branches as $$h\to0$$ were achieved when $$\mathcal{F}$$ is Riemannian and $$g$$ bundle-like ($$g_H$$ is “leafwise constant”). For instance, given a Schwartz function $$f$$ on $$\mathbb{R}$$, a version of the semiclassical Weil formula was proved by the first author in [Math. Ann. 313, 763–783 (1999; Zbl 0930.58017], describing the asymptotics of $$f(\Delta_h)$$ as $$h\to0$$ when $$\mathcal{F}$$ is Riemannian and $$g$$ bundle-like. Under these conditions, it was also shown by the first author and the reviewer [Geom. Funct. Anal. 10, 977–1027 (2000; Zbl 0965.57024] that the spectral branches converging to zero as $$h\to0$$, called small spectrum, are related to the spectral sequence $$E_k$$ of $$\mathcal{F}$$, which is the obvious generalization of the Leray spectral sequence of fiber bundles defined by filtering the de Rham complex.
The paper under review is a presentation of the above results and other recent results whose detailed proofs are in two preprints of the second author and his publication [Sb. Math. 199, No. 2, 307–318 (2008); translation from Mat. Sb. 199, No. 2, 149–160 (2008; Zbl 1157.58009)]. These new results provide very concrete computations of the spectral branches in two examples. In the first one, $$M$$ is the $$2$$-torus $$\mathbb{R}^2/\mathbb{Z}^2$$, $$\mathcal{F}$$ is the Kronecker flow defined by lines of a fixed slope $$\alpha$$ on $$\mathbb{R}^2$$, and $$g$$ is the flat metric. In particular, when $$\alpha\not\in\mathbb{Q}$$, it is shown that the spectral distribution function $$N_h(\lambda)$$ of $$\Delta_h$$ on functions has the following asymptotic expression when $$h\to0$$: $N_h(\lambda)=\frac{1}{4\pi}h^{-1}\lambda+o(h^{-1})\;.$ The second example is very interesting because the foliation is not Riemannian: $$(M,g)$$ is a three-dimensional Riemannian Heisenberg manifold $$\Gamma\backslash H$$, and $$\mathcal{F}$$ is given by a left invariant vector field on $$H$$ which is not in the center of its Lie algebra. In this case, the computation of the spectrum of the Laplacian on functions was made by C. S. Gordon and E. N. Wilson [Mich. Math. J. 33, 253–271 (1986; Zbl 0599.53038)]. The authors compute the spectrum also on forms, obtaining the following asymptotic formula as $$h\to 0$$ for the trace of the heat operator: $\text{tr}e^{-t\Delta_h}=\frac{h^{-2}}{4\pi t}\int_{-\infty}^{+\infty}\frac{\eta}{\sinh(t\eta)}\,e^{-t\eta^2}\,d\eta+o(h^{-2})\;.$ Here, the factor $$\frac{\eta}{\sinh(t\eta)}$$ is given by the leafwise variation of $$g_H$$; thus this factor would not show up in the case of a Riemannian foliation. Nevertheless it is shown that the small spectrum of this example is related to the spectral sequence $$E_k$$ like in the case of Riemannian foliations.
For the entire collection see [Zbl 1134.58002].

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 57R30 Foliations in differential topology; geometric theory
Full Text:


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# [IPython-dev] IPython Notebook + MathJax + Firefox

Shriramana Sharma samjnaa@gmail....
Tue Jun 18 12:07:15 CDT 2013

Hello people. I hope this is the right forum for this.

First I thank you very much for this great software -- you make my
Python usage much more pleasant. I use IPy3 0.13.2 with Python 3.3.1
on Kubuntu Raring.

Normally I use the Qt Console interface but sometimes I do open up the
notebook, and I find that the OTF MathJax fonts don't get loaded on
Firefox because of a Firefox security "feature" that prevents fonts
from the location that calls them.


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# ECE 417 MP3: Face Detection using Faster RCNN¶

This MP is an implementation of the Faster RCNN object detector, to perform face detection using a small dataset extracted from the very large WIDER FACE dataset.

• When you download and unzip that package, you will find a file called mp3_overview.ipynb. If you open that file using any ipython or Jupyter notebook, you will see a live version of the web page you are looking at right now.
• The only code that you will submit is the file mp3.py. The parts that you need to fill in are marked with NotImplementedErrors.
• As you edit mp3.py, you can see the results of your edits by running the corresponding code blocks in this file. Notice that each relevant code block starts with a importlib.reload(mp3), which will reload your most recent edited version, and run it.
• When everything on this page seems to run OK, you can run the unit tests by typing python run_tests.py or python run_tests.py -j in a console.
• When the unit tests all succeed on your own machine, you can try them on the autograder. Just upload the file mp3.py (if you upload other files, they will not be graded).

The rest of this file contains code for debugging mp3.py. If you are looking at this file on the web, you can browse the blocks below, to see what kind of material the MP will cover. If you are interacting with this file on your own machine, you can run the code yourself, to test your own version of mp3.py, and you can edit these code blocks as much as you want, in order to print out the variables, or any other debugging step that's useful to you.

In [559]:
import numpy  as np
import matplotlib.pyplot as plt
%matplotlib inline
import matplotlib.figure
import importlib, os, sys, h5py

This block imports your mp3.py code:

In [560]:
import mp3

## 0. Browsing the Data¶

First, we'll create a function that plots face rectangles on top of the image. Then we'll use it to look at some of the data.

In [561]:
def show_image_with_rectangles(image,rects):
fig=plt.figure(figsize=(7,7))
plt.imshow(image)
for rect in rects:
x = rect[0]+np.array([-rect[2]/2,rect[2]/2])
y = rect[1]+np.array([-rect[3]/2,rect[3]/2])
plt.plot(x,y[0]*np.ones(2),'c-',linewidth=3)
plt.plot(x,y[1]*np.ones(2),'c-',linewidth=3)
plt.plot(x[0]*np.ones(2),y,'c-',linewidth=3)
plt.plot(x[1]*np.ones(2),y,'c-',linewidth=3)

It's good practice to use "importlib.reload" before you call anything from mp3.py. That way, every time you make changes to that file, your changes will be drawn in to this notebook page immediately.

This chunk of code loads in the mp3_dataset object. We'll load in the first datum, and find out what members it contains.

In [562]:
importlib.reload(mp3)
mp3_dataset = mp3.MP3_Dataset('data')
datum=mp3_dataset[0]
for k in datum.keys():
print('Object %s has shape %s'%(k,datum[k].shape))

Object image has shape (224, 224, 3)
Object features has shape (1, 512, 14, 14)
Object rects has shape (3, 10)
Object target has shape (9, 196, 5)

Next, let's plot the image, and overlay its reference rectangles on top:

In [563]:
show_image_with_rectangles(datum['image'],datum['rects'])

Next, let's plot the anchor rectangles. For reasons of efficiency, the anchors are stored in the opposite order: anchors[i,xy,a] is the i'th term in the rectangle corresponding to the a'tjh anchor (0<=a<=8) associated with the xy'th image position (0<=xy<=195). In order to make it easier to plot, let's create a transposed version.

In [564]:
anchor_rects = np.transpose(mp3_dataset.anchors,[1,2,0])
[ mp3_dataset.anchors.shape, anchor_rects.shape ]

Out[564]:
[(4, 196, 9), (196, 9, 4)]

In order to plot the smallest square at each anchor position, we just need to choose the zero'th anchor associated with each position.

In [565]:
show_image_with_rectangles(datum['image'],anchor_rects[:,0,:])

To better understand the anchors, let's show all of the anchors associated with a position somewhere out in the middle of the image -- say, position number x=7, y=7 (thus xy = 7x14+7-1=104).

In [566]:
show_image_with_rectangles(datum['image'],anchor_rects[104,:,:])

Finally: the classification targets and regression targets are encoded into the array called datum['target']. Specifically, the binary classification target for the a'th anchor at the i'th position is encoded as datum['target'][a,i,4], and if the classification target is 1, then the regression target is encoded as datum['target'][a,i,0:4].

Notice the min(np.log(2),regression_target) here. That keeps the computed rectangle from ever being more than twice as large as the anchor rectangle. I recommend that you use such a limit in your code, because sometimes the neural net outputs get very large.

In [567]:
def target2rect(regression_target, anchor):
rect = np.zeros(4)
rect[0] = regression_target[0]*anchor[2]+anchor[0]
rect[1] = regression_target[1]*anchor[3]+anchor[1]
rect[2] = np.exp(min(np.log(2),regression_target[2]))*anchor[2]
rect[3] = np.exp(min(np.log(2),regression_target[3]))*anchor[3]
return(rect)

The Faster-RCNN coding scheme creates a target whenever a reference rectangle and an anchor rectangle have an IoU (intersection-over-union) greater than 0.7. For that reason, there are a lot more target rectangles than there were reference rectangles:

In [568]:
target_rects = []
for a in range(9):
for i in range(196):
if datum['target'][a,i,4]==1:
rect = target2rect(datum['target'][a,i,0:4],anchor_rects[i,a,:])
target_rects.append(rect)
np.array(target_rects).shape

Out[568]:
(416, 4)

If we plot the target rectangles, though, we should see that the number of distinct target rectangles is just the same as the number of reference rectangles. All of the extra targets are just duplicates of the same original references:

In [569]:
show_image_with_rectangles(datum['image'],target_rects)


text:
Cplex is stuck after root node relaxation solution

I am solving an MIP through Benders decomposition (coded both generic and legacy callback versions), by employing Java with Cplex 12.9. For some of the instances, Cplex is stuck for two hours (time limit is two hours) after solving root node relaxation very fast (in less than a second) and finally at the end of two hours it reports a single line of branch and bound tree (root node) and exits due to time limit. Nothing is reported after root node relaxation solution and before the first line of branch and bound tree (to understand what Cplex is doing for two hours). I turned off reductions and presolve and all possible cuts. Nothing changed. Please see that portion of the code:

master.setParam(IloCplex.IntParam.MIPDisplay, 4);
master.setParam(IloCplex.Param.Threads, nThreads);
master.setParam(IloCplex.Param.Preprocessing.Reduce, 0);
master.setParam(IloCplex.BooleanParam.PreInd, false);
master.setParam(IloCplex.IntParam.MIPEmphasis, 2);
master.setParam(IloCplex.IntParam.VarSel, 3);
master.setParam(IloCplex.Param.MIP.Strategy.HeuristicFreq, -1);
master.setParam(IloCplex.DoubleParam.TiLim, 7200);

master.setParam(IloCplex.IntParam.Cliques, -1);
master.setParam(IloCplex.IntParam.Covers, -1);
master.setParam(IloCplex.IntParam.DisjCuts, -1);
master.setParam(IloCplex.IntParam.FlowCovers, -1);
master.setParam(IloCplex.IntParam.FracCuts, -1);
master.setParam(IloCplex.IntParam.GUBCovers, -1);
master.setParam(IloCplex.IntParam.LiftProjCuts, -1);
master.setParam(IloCplex.IntParam.MCFCuts, -1);
master.setParam(IloCplex.IntParam.MIRCuts, -1);
master.setParam(IloCplex.IntParam.FlowPaths, -1);
master.setParam(IloCplex.IntParam.ZeroHalfCuts, -1);

master.setParam(IloCplex.Param.MIP.Cuts.BQP, -1);
master.setParam(IloCplex.Param.MIP.Cuts.Cliques, -1);
master.setParam(IloCplex.Param.MIP.Cuts.Covers, -1);
master.setParam(IloCplex.Param.MIP.Cuts.Disjunctive, -1);
master.setParam(IloCplex.Param.MIP.Cuts.FlowCovers, -1);
master.setParam(IloCplex.Param.MIP.Cuts.Gomory, -1);
master.setParam(IloCplex.Param.MIP.Cuts.GUBCovers, -1);
master.setParam(IloCplex.Param.MIP.Cuts.Implied, -1);
master.setParam(IloCplex.Param.MIP.Cuts.LiftProj, -1);
master.setParam(IloCplex.Param.MIP.Cuts.LocalImplied, -1);
master.setParam(IloCplex.Param.MIP.Cuts.MCFCut, -1);
master.setParam(IloCplex.Param.MIP.Cuts.MIRCut, -1);
master.setParam(IloCplex.Param.MIP.Cuts.PathCut, -1);
master.setParam(IloCplex.Param.MIP.Cuts.RLT, -1);
master.setParam(IloCplex.Param.MIP.Cuts.ZeroHalfCut, -1);

Please see the log file below:

CPXPARAM_Preprocessing_Presolve                  0
CPXPARAM_Preprocessing_Reduce                    0
CPXPARAM_Threads                                 40
CPXPARAM_MIP_Cuts_Cliques                        -1
CPXPARAM_MIP_Cuts_Covers                         -1
CPXPARAM_MIP_Display                             4
CPXPARAM_MIP_Strategy_VariableSelect             3
CPXPARAM_MIP_Cuts_FlowCovers                     -1
CPXPARAM_MIP_Cuts_Implied                        -1
CPXPARAM_MIP_Cuts_GUBCovers                      -1
CPXPARAM_MIP_Cuts_Gomory                         -1
CPXPARAM_MIP_Cuts_PathCut                        -1
CPXPARAM_MIP_Cuts_MIRCut                         -1
CPXPARAM_MIP_Cuts_Disjunctive                    -1
CPXPARAM_Emphasis_MIP                            2
CPXPARAM_MIP_Cuts_ZeroHalfCut                    -1
CPXPARAM_MIP_Cuts_MCFCut                         -1
CPXPARAM_MIP_Cuts_LiftProj                       -1
CPXPARAM_MIP_Cuts_LocalImplied                   -1
CPXPARAM_MIP_Cuts_BQP                            -1
CPXPARAM_MIP_Cuts_RLT                            -1
CPXPARAM_MIP_Strategy_HeuristicFreq              -1
CPXPARAM_TimeLimit                               7200
Lazy constraint(s) or lazy constraint callback is present.
Disabling non-linear reductions (CPX_PARAM_PRELINEAR) in presolve.
Clique table members: 114063.
MIP emphasis: optimality.
MIP search method: dynamic search.
Parallel mode: deterministic, using up to 40 threads.
Parallel mode: deterministic, using up to 40 threads for concurrent optimization.
Initializing dual steep norms . . .
Initializing dual steep norms . . .

Dual simplex solved model.

Root relaxation solution time = 0.61 sec. (227.78 ticks)

Nodes                                         Cuts/
Node  Left     Objective  IInf  Best Integer    Best Bound    ItCnt     Gap

0     0  3898607.0000     0                      0.0000        0

User cuts applied:  2

Root node processing (before b&c):
Real time             = 7199.43 sec. (2190540.20 ticks)
Parallel b&c, 40 threads:
Real time             =    0.00 sec. (0.00 ticks)
Sync time (average)   =    0.00 sec.
Wait time (average)   =    0.00 sec.
------------
Total (root+branch&cut) = 7199.43 sec. (2190540.20 ticks)

Thank you.

• Can you verify that the cuts that you are adding violate the current solution? I have seen cases where the cut does not violate, such that you get the same solution in the next iteration, after which you add the same cut again. Such an infinite loop could explain the behaviour you describe. – Kevin Dalmeijer Dec 22 '19 at 14:37
• I am adding three types of cuts. 1. Benders optimality cuts added at integer solutions, 2. Benders optimality cuts added at fractional solutions, and 3. Valid inequalities added only at the root node relaxation. To eliminate the possibility of a loop I am just using 1st type of cuts. And I am pretty sure that Benders optimality cuts are cutting off the current solution. As you can see there are only two Benders cuts added in the log file. Thank you. – Vedat BAYRAM Dec 22 '19 at 15:18
• Was this output from the legacy callback version or the generic callback version? For debugging purposes, you might set the node log frequency to 1 (so that a line is printed for every node). Also, you might want to print a time stamp each time the callback is entered and exited (also printing the thread #, to keep them straight). – prubin Dec 22 '19 at 21:41
• Thank you for the comment. Log frequency is set to 4, in which case Cplex is supposed to "Display the number of cuts added since previous display; information about the processing of each successful MIP start; elapsed time in seconds and elapsed time in deterministic ticks for integer feasible solutions and information about the LP subproblem at root" as defined by IBM support. I am now trying your advice regarding time stamp at each callback and thread id. Thank you. – Vedat BAYRAM Dec 23 '19 at 15:13
• I am using generic callback now. I checked the time stamps and thread Ids. Additionally after root node relaxation and before the single line of branch and bound tree, now I have: Time: 1.446242725 Thread Id: 0 Time: 3.146363022 Thread Id: 0 in Candidate >>> Adding optimality cut at int point: Time: 1926.164073447 Thread Id: 0 in Candidate >>> Adding optimality cut at int point: Time: 7201.699062 Thread Id: 0 – Vedat BAYRAM Dec 23 '19 at 18:24


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# Matrix of a linear transformation for an integral?

by marathon
Tags: integral, linear, matrix, transformation
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,691 Is there anything in your book about itegrals as linear transformations? Or about writing a general linear transformation as a matrix? Those are what you need here. An integral is a linear transformation: $\int af(x)+ bg(x)dx= a\int f(x)dx+ b\int g(x)dx$. To write a linear transformation from vector space U to vector space V, given ordered bases for each, do the following. Apply the linear transformation to the first vector in the ordered basis for U. That will be in V so can be written as a linear combination of the ordered basis for V. The coefficients of that linear combination will be the first column in the matrix. Do the same with the second vector in the ordered basis for U to get the second column, etc.


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TA

# Reduce the equation to one of the standard forms, classify the surface, and sketch it.$4x^2 - y + 2z^2 = 0$

## elliptic paraboloid

Vectors

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### Video Transcript

All right. We want to write the standard form of access square, man. That's why Has to this curricula zero. So then we just move right to the other side. So I get why he was 24 X squared plus two G square. This is this is the standard form and this is an elliptic parabola. It and the reason is the following. So if I see that, why cannot be negative? So this is really on the I want you to start up boy and that for any value of Y or any Why value we call this value. Okay, what we get is that four X squared plus two G square equals two K. Is an ellipse on the X G plan. Mm hmm. So this means that at every value of K we get ellipse and the value of cable increases the the dimension population increases. So, so what we get is a parabola. Well, joining these ellipses and let me draw this for the sake of convenience. Again, let me draw the Y axis on the top. And this is my ex cheap. So like I said, why value is always positive? So this is a positive value and or what we call the zero, my X. And year zero. Then from what I got one I get I instead of a circle, I get an ellipse like this. Similarly for why equal to two, I get a bigger lips And what I call the three I got a bigger bigger election. So so if I join all of this, so you get something like, so this is an elliptic parabolas. So each if you if you draw an intersection, if you draw a plane here, this plane intersects this elliptic parabola. It's what we get in the intersection is an ellipse. This is exactly what is up.

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# How To Solve Base 15 Equations

Question: "Find all values of the digit m, if any, for which $m{\rm A} \mbox{ base fifteen} + m{\rm B} \mbox{ base fifteen} + m{\rm E} \mbox{ base fifteen} − \mbox{CD base fifteen} = 304 \mbox{ base } m.$"

To start off, I converted CD to base ten and got $193.$ I then converted $304$ base $m$ to base ten and got $3m^2+0m+4.$ I moved the $193$ over to the other side of the equation and got $m{\rm A} \mbox{ base fifteen} + m{\rm B} \mbox{ base fifteen} + m{\rm E} \mbox{ base fifteen} = 193m^2-189.$

Next, I added $m\rm A,$ $m\rm B,$ and $m\rm E$ to get the number "$(3m+2)5$" base fifteen, which I then converted to base ten and set equal to $193m^2-189.$ Then, I solved, but didn't get an integer as a result.

Could someone confirm if what I did was right / wrong?

Thanks (and sorry about the tags or lack thereof - bases wasn't one of the options and it didn't let me create a tag for it).

## 1 Answer

You made a mistake when you "moved the 193 over". You should still have a $3m^2$ term. Also, it's best to convert everything to base ten as soon as possible. This yields: \begin{align*} (15m + 10) + (15m + 11) + (15m + 14) - (15 \cdot 12 + 13) &= 3m^2 + 4 \\ 45m - 158 &= 3m^2 + 4 \\ 0 &= 3m^2 - 45m + 162 \\ 0 &= m^2 - 15m + 54 \\ 0 &= (m - 6)(m - 9) \\ m &= 6,9 \end{align*}

• Sorry, that was my mistake when copying over to stack exchange. I still had am m^3 term, but I see I went wrong somewhere else along the line. Thanks for pointing that out! – Math Student Oct 27 '14 at 21:45


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# Items tagged with rotaterotate Tagged Items Feed

### 3-D Cube in Maple

February 24 2013 by Maple

1

5

Hello,

I am trying to make a three dimensional cube in maple by using matrices. I also have to make it rotate about the x,y,z axes, project it with three different origins, and translate it (I'm assuming this means translate it across an axis?). I have already made the cube, but when I try to rotate it across one of the axes by multiplying my matrix by one of the rotation matrices (this is what I put un maple: P1:= matrix([[1,0,0,0],[0,cos(theta),-sin(theta),0],[0,sin(theta),cos(theta),0...

### Plotting a filled semicircle rotated...

December 01 2012 by Maple

1

2

I can't solve probably very easy problem. How to plot a filled semicircle which is rotated across one axis by an angle alpha? I came to the solution which I don't consider as the best one (since e.g. it will not work for alpha=Pi/2*(odd integer)) and I believe someone of you will show me a better approach. Thank you in advance.

My solution:

alpha := (1/6)*Pi:plot3d([x, y, y*tan(alpha)], x = -1 .. 1, y = 0 .. sqrt(1-x^2)*cos(alpha), axes = normal, labels = ["x", "y", "z"...

### How to plot S-shape line

May 06 2012 by Maple

0

0

Hello to everyone

Pls help,urgent....

I need to draw the deformation of one story building (for simplicity, 4 columns and 2 floors) whose columns and floors  under influence of external load acting to the left of the building at its intersection point undergo the deform shape like S . So means that when the building moves to the right its intersection point of column and floor has some small rotation.

I am new to Maple environment...

### animation of 3D plot rotation

January 24 2012 by Maple 15

8

6

Suppose that you wish to animate the whole view of a plot. By whole view, I mean that it includes the axes and is not just a rotation of a plotted object such as a surface.

One simple way to do this is to call plots:-animate (or plots:-display on a list of plots supplied in a list, with its insequence=true option). The option orientation would contain the parameter that governs the animation (or generates the sequence).

But that entails recreating the same plot each time. The plot data might not even change. The key thing that changes is the ORIENTATION() descriptor within each 3d plot object in the reulting data structure. So this is inefficient in two key ways, in the worst case scenario.

1) It may even compute the plot's numeric results, as many times as there are frames in the resulting animation.

2) It stores as many instances of the grid of computed numeric data as there are frames.

We'd like to do better, if possible, reducing down to a single computation of the data, and a single instance of storage of a grid of data.

To keep this understandable, I'll consider the simple case of plotting a single 3d surface. More complicated cases can be handled with revisions to the techniques.

Avoiding problem 1) can be done in more than one way. Instead of plotting an expression, a procedure could be plotted, where that procedure has option remember so that it automatically stores computed results an immediately returns precomputed stored result when the arguments (x and y values) have been used already.

Another way to avoid problem 1) is to generate the unrotated plot once, and then to use plottools:-rotate to generate the other grids without necessitating recomputation of the surface. But this rotates only objects in the plot, and does alter the view of the axes.

But both 1) and 2) can be solved together by simply re-using the grid of computed data from an initial plot3d call, and then constructing each frame's plot data structure component "manually". The only thing that has to change, in each, is the ORIENTATION(...) subobject.

At 300 frames, the difference in the following example (Intel i7, Windows 7 Pro 64bit, Maple 15.01) is a 10-fold speedup and a seven-fold reduction is memory allocation, for the creation of the animation structure. I'm not inlining all the plots into this post, as they all look the same.

restart:
P:=1+x+1*x^2-1*y+1*y^2+1*x*y:

st,ba:=time(),kernelopts(bytesalloc):

plots:-animate(plot3d,[P,x=-5..5,y=-5..5,orientation=[A,45,45],
axes=normal,labels=[x,y,z]],
A=0..360,frames=300);

time()-st,kernelopts(bytesalloc)-ba;

1.217, 25685408

restart:
P:=1+x+1*x^2-1*y+1*y^2+1*x*y:

st,ba:=time(),kernelopts(bytesalloc):

g:=plot3d(P,x=-5..5,y=-5..5,orientation=[-47,666,-47],
axes=normal,labels=[x,y,z]):

plots:-display([seq(PLOT3D(GRID(op([1,1..2],g),op([1,3],g)),
remove(type,[op(g)],
specfunc(anything,{GRID,ORIENTATION}))[],
ORIENTATION(A,45,45)),
A=0..360,360.0/300)],
insequence=true);

time()-st,kernelopts(bytesalloc)-ba;

0.125, 3538296

By creating the entire animation data structure manually, we can get a further factor of 3 improvement in speed and a further factor of 3 reduction in memory allocation.

restart:
P:=1+x+1*x^2-1*y+1*y^2+1*x*y:

st,ba:=time(),kernelopts(bytesalloc):

g:=plot3d(P,x=-5..5,y=-5..5,orientation=[-47,666,-47],
axes=normal,labels=[x,y,z]):

PLOT3D(ANIMATE(seq([GRID(op([1,1..2],g),op([1,3],g)),
remove(type,[op(g)],
specfunc(anything,{GRID,ORIENTATION}))[],
ORIENTATION(A,45,45)],
A=0..360,360.0/300)));

time()-st,kernelopts(bytesalloc)-ba;

0.046, 1179432

Unfortunately, control over the orientation is missing from Plot Components, otherwise such an "animation" could be programmed into a Button. That might be a nice functionality improvement, although it wouldn't be very nice unless accompanied by a way to export all a Plot Component's views to GIF (or mpeg!).

The above example produces animations each of 300 frames. Here's a 60-frame version:

### 3D plot rotation efficiency

October 12 2011 by Maple

4

0

I was recently looking at rotating a 3D plot, using plottools:-rotate, and noticed something inefficient.

In the past few releases of Maple, efficient float[8] datatype rtables (Arrays or hfarrays) can be used inside the plot data structure. This can save time and memory, both in terms of the users' creation and manipulation of them as well as in terms of the GUI's ability to use them for graphic rendering.

What I noticed is that, if one starts with a 3D plot data structure containing a float[8] Array in the MESH portion, then following application of plottools:-rotate a much less efficient list-of-lists is produced in the resulting structure.

Likewise, an effiecient float[8] Array or hfarray in the GRID portion of a 3D plot structure gets transformed by plottools:-rotate into an inefficient list-of-lists object in the MESH portion of the result. For example,

restart:

p:=plot3d(sin(x),x=-6..6,y=-6..6,numpoints=5000,style=patchnogrid,
axes=box,labels=[x,y,z],view=[-6..6,-6..6,-6..6]):

seq(whattype(op(3,zz)), zz in indets(p,specfunc(anything,GRID)));
hfarray

pnew:=plottools:-rotate(p,Pi/3,0,0):

seq(whattype(op(1,zz)), zz in indets(pnew,specfunc(anything,MESH)));
list

The efficiency concern is not just a matter of the occupying space in memory. It also relates to the optimal attainable methods for subsequent manipulation of the data.

It may be nice and convenient for plottools to get as much mileage as it can out of plottools:-transform, internally. But it's suboptimal. And plotting is a topic where dedicated, optimized helper routines for some particular data format is justified and of merit. If we want plot manipulation to be fast, then both Library-side as well as GUI-side operations need more case-by-case-optimizated.

Here's an illustrative worksheet, using and comparing memory performance with a (new, alternative) procedure that does inplace rotation of a 3D MESH. plot3drotate.mw

### batch files to crop and rotate pdf image...

September 29 2011 by Maple

2

0

If you have been logged in at mapleprimes in the last week you will know that I'm currently going through an obsession with plots. Indeed, some deadline is looming and it is all very stressful. For reference, and hopefully it may be of use to someone somewhere someday, I produce 2D plots with Standard GUI in the postscript format using plotsetup(ps). I'm on Windows for this. (3D plots aren't so hot with Standard GUI)

I use LaTeX for my documents. I used to insert postscript...

 Page 1 of 1


text:
## Algebra and Trigonometry 10th Edition

Let x be one natural number. x+1 is the other. Hence, $\frac{1}{x} - \frac{1}{x+1} = \frac{1}{4}*\frac{1}{x}$ Multiply by x(x+1) on both sides (x+1) - x = (x+1)/4 x+1 = 4 x=3 x+1=4


text:
Board index Computation of Integrals Arc Length of the Sine Function

## Arc Length of the Sine Function

Post your questions related to Computation of Integrals here.

Moderators: galactus, Random Variable, sos440

### Arc Length of the Sine Function

Tue Nov 29, 2016 6:17 pm

Posts: 10
On the Wikipedia page for the sine function the arc length for a full period of $\sin(x)$ is given as

I'm trying to derive this result and so far I have gotten to the integral

$$L=4\int_0^1\sqrt{\frac{1+t^2}{1-t^2}}dt.$$

I'm aware that this is a complete elliptic integral of the second kind and that $\frac{\Gamma^2\left(1/4\right)}{\sqrt{2\pi}}$ is the Lemniscate Constant, but that is really all I know. My guesses of how to continue so far are to try to transform the integral into lemniscatic integrals and apply the beta function. I'm stuck in a rut because I don't know any identities that can help me go on, so if you guys have any hints, substitutions, transformations, or identities to share that would be great.
The internet is a $\sum^{\infty}_{n=0}$tube$_n$ filled with cats!

### Re: Arc Length of the Sine Function

Tue Nov 29, 2016 10:39 pm

Posts: 27
The substution $u = t^4$ transforms the integral into:

$\displaystyle \large \int_0^1 \frac{u^{-\frac{3}{4}} + u^{-\frac{1}{4}}}{\sqrt{1-u}} \, \text{d}u$

The standard definition of the Beta Function can be applied immediately to yield the desired result.

### Re: Arc Length of the Sine Function

Tue Nov 29, 2016 10:42 pm

Posts: 10
The substution $u = t^4$ transforms the integral into:

$\displaystyle \large \int_0^1 \frac{u^{-\frac{3}{4}} + u^{-\frac{1}{4}}}{\sqrt{1-u}} \, \text{d}u$

The standard definition of the Beta Function can be applied immediately to yield the desired result.

Thankyou! I can't believe I didn't see that! I had tried $u=t^2$ and didn't get anywhere.
The internet is a $\sum^{\infty}_{n=0}$tube$_n$ filled with cats!