Question

Determine whether F is the gradient of some function f. If it is, find such a function f.

1. F(x,y)=(e^y)i+((xe^y)+y)j

2. F(x,y)=(2xyz)i+(x^2*z)j+(x^2*y+1)k

Answer 1

if its a gradient , it should satisfy:
\[\frac{\partial e^y}{\partial y}=\frac{\partial (xe^y +y)}{\partial x}\]
which they do

Answer 2

Gotcha. So what are the steps/formula for finding the function?

Answer 3

Let's say its gradient of some function G

Answer 4

Then \[\frac{\partial G}{\partial x}=e^y\]

Answer 5

Integrating respect to x:
\[G=xe^y +g(y)\]
differentiating this respect to y:
\[\frac{\partial G}{\partial y} =xe^y+g'(y)\]

Answer 6

but dG/dy= xe^y)+y
so
\[xe^y+g'(y)=xe^y+y\]

Answer 7

from here you see that g'(y)=y

Answer 8

so integrating:
g(y) = 1/2y^2
so
G(x,y)=xe^y+1/2y^2

Answer 9

got it?

Answer 10

Hmmm, can you write out your steps in words? I understand to a degree. I just wanna write this out on paper so I can recheck the steps.

Answer 11

I did write it in words

Answer 12

you can write this on paper like this and you will be fine

Answer 13

more clear impossible

Answer 14

Alright I think I got it then, thanks.

Answer 15

yw