A few clarifications on Utility equations and indifference curves? I have the utility equation $U(a,b) = a^{2}b^{3}$

How can I tell if the indifference curves are convex?
I was under the impression that if:

$U_{a} &gt; 0$  and $U_{b} &lt; 0$  

then the curve would be convex. In this case, those conditions don't hold, but when I try graphing the equation, the curve appears convex. What am I missing?

Also are these conditions true:

if $U_{a} &gt; 0$ , there is not diminishing marginal utility.
if &lt; 0 , there is diminishing marginal utility,
and if = 0 , constant. 

Are these conditions the same for good $b$ ?

I've been given lots of conflicting info and am now just confused.
From a mathematical point of view, the indifference curve is an equation

$$I_{\bar u}:a^{2}b^{3} = \bar u$$ 

for every fixed $\bar u$. Put $b$ on the vertical axis and $a$ on the horizontal axis and write this as a function of $a$:

$$b^3 = \bar ua^{-2} \implies b = (\bar u)^{1/3} a^{-2/3}$$

This fully characterizes the indifference curve for $\bar u$. Now you have a one-dimensional function, and for it to be convex, its second derivative with respect to $a$ must be non-negative. In fact it is strictly positive (we assume goods are measured in positive quantities, and we ignore the uninteresting corner solution for $\bar u=0$ which is a single point).