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"Everyone knows how to get the derivative of a function purely mechanically. Many people understand the physical meaning of the derivative: it shows how fast or slow the function changes at a given point. This is very important in mathematics, physics, and other fields where you need to understand how one quantity changes in relation to another.

Everyone knows that the transposition of a matrix is the rearranging of rows and columns, and it is also needed to calculate the determinant or to reveal the symmetry of a matrix. Everyone knows how to transpose a matrix mechanically to get a more convenient representation for solving systems of equations, etc. But few realize that the physical meaning of transposing a matrix ... "