I have a function:
$e(p_1,p_2,u) = \frac{p_1p_2u^2}{4(4p_1+p_2)}$ and I'm being asked to calculate the gradient vector with respect to p. That is, I want to find:
$∇_pe(p_1,p_2,u)$
I understand the concept of the gradient with respect to one variable, I take the derivative of each. But now I'm taking the derivative with respect to a vector? Do I sum up the components of each derivative or?
An example, or a solution, would be great.
The gradient function does create a vector from a scalar it operates on a scalar function to return a vector.
The Grad function $$\nabla$$ can be seen as a bracket that is distributed to the function e.
$$\nabla = \frac{\partial}{\partial x} \hat{i} + \frac{\partial}{\partial y} \hat{j} + \frac{\partial}{\partial z} \hat{k}$$
thus
$$\nabla_{p} e(p_{1},p_{2},u) = \frac{\partial e(p_{1},p_{2},u)}{\partial p_{1}} \hat{i} + \frac{\partial e(p_{1},p_{2},u)}{\partial p_{2}} \hat{j} + \frac{\partial e(p_{1},p_{2},u)}{\partial u} \hat{k}$$
it to be done.
If this is for optimization in economics perhaps these vectors would be applied to a markov chain