Calculating Divergence of a Vector Field in Three Dimensions

[I_laff]

If I have a vector field say $v = e^{z}(y\hat{i}+x\hat{j})$, and I want to calculate the divergence. Do I only take partial derivatives with respect to x and y (like so, $\frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y}$) or should I take partial derivatives with respect to x, y and z (like so, $\frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}$). I'm confused as to which one because there is no $\hat{k}$ unit vector, but z is changing and the graph should therefore be three dimensional.

 

[jedishrfu]

Where did you get this problem? It could be a typo. Did you check the online errata page associated with the book?

It seems the e^z term is the k-component in which case you take its partial derivative with respect to z.

 

[I_laff]

The example was made up however, I remember seeing a question like this and it had me confused.

 

[I_laff]

Oh apologies I made an error in my vector field equation, it has been corrected now.

 

[jedishrfu]

Okay so the k-component would be $0\hat{k}$

Now take your x,y,z partials for the divergence.

 

[I_laff]

So since the k component is 0, would that not mean that the divergence is calculated using only the partials of x and y since $A_z = 0$.

 

[jedishrfu]

Yes but you need to remember to do all of them.