Generating a table of gradients

Question

I have some function (say, $f(x,y)=x^2+y$) I want to take the gradient of. I then want to make a table of $|\nabla f|$ evaluated at different values of $x$ and $y$. Here's the code I wrote:

Table[Norm[Grad[x^2+y,{x,y}]],{x,1,10},{y,1,10}]

I'm getting the following error message:

Grad::nocoord: {1,1} is not a non-empty list of valid variables. $\gg$

Grad::nocoord: {1,2} is not a non-empty list of valid variables. $\gg$

Grad::nocoord: {1,3} is not a non-empty list of valid variables. $\gg$

General::stop: Further output of Grad::nocoord will be suppressed during this calculation. $\gg$

What am I doing wrong?

Answer 1

Table[Evaluate[Norm[Grad[x^2 + y, {x, y}]]], {x, 1, 10}, {y, 1, 10}]

or

Table[Norm[Grad[x^2 + y, {x, y}]] /. {x -> u, y -> v}, {u, 1, 10}, {v, 1, 10}]

both give (when wrapped with TeXForm):

$\left( \begin{array}{cccccccccc} \sqrt{5} & \sqrt{5} & \sqrt{5} & \sqrt{5} & \sqrt{5} & \sqrt{5} & \sqrt{5} & \sqrt{5} & \sqrt{5} & \sqrt{5} \\ \sqrt{17} & \sqrt{17} & \sqrt{17} & \sqrt{17} & \sqrt{17} & \sqrt{17} & \sqrt{17} & \sqrt{17} & \sqrt{17} & \sqrt{17} \\ \sqrt{37} & \sqrt{37} & \sqrt{37} & \sqrt{37} & \sqrt{37} & \sqrt{37} & \sqrt{37} & \sqrt{37} & \sqrt{37} & \sqrt{37} \\ \sqrt{65} & \sqrt{65} & \sqrt{65} & \sqrt{65} & \sqrt{65} & \sqrt{65} & \sqrt{65} & \sqrt{65} & \sqrt{65} & \sqrt{65} \\ \sqrt{101} & \sqrt{101} & \sqrt{101} & \sqrt{101} & \sqrt{101} & \sqrt{101} & \sqrt{101} & \sqrt{101} & \sqrt{101} & \sqrt{101} \\ \sqrt{145} & \sqrt{145} & \sqrt{145} & \sqrt{145} & \sqrt{145} & \sqrt{145} & \sqrt{145} & \sqrt{145} & \sqrt{145} & \sqrt{145} \\ \sqrt{197} & \sqrt{197} & \sqrt{197} & \sqrt{197} & \sqrt{197} & \sqrt{197} & \sqrt{197} & \sqrt{197} & \sqrt{197} & \sqrt{197} \\ \sqrt{257} & \sqrt{257} & \sqrt{257} & \sqrt{257} & \sqrt{257} & \sqrt{257} & \sqrt{257} & \sqrt{257} & \sqrt{257} & \sqrt{257} \\ 5 \sqrt{13} & 5 \sqrt{13} & 5 \sqrt{13} & 5 \sqrt{13} & 5 \sqrt{13} & 5 \sqrt{13} & 5 \sqrt{13} & 5 \sqrt{13} & 5 \sqrt{13} & 5 \sqrt{13} \\ \sqrt{401} & \sqrt{401} & \sqrt{401} & \sqrt{401} & \sqrt{401} & \sqrt{401} & \sqrt{401} & \sqrt{401} & \sqrt{401} & \sqrt{401} \\ \end{array} \right)$