If

F(x,y,z) = sinyi + (xcosy + z )j + yk

find a function f (x,y,z) such that

F = grad f

how would you go about solving this one?

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- Mar 18th 2011, 04:05 PMheatlyVector Field
If

F(x,y,z) = sinyi + (xcosy + z )j + yk

find a function f (x,y,z) such that

F = grad f

how would you go about solving this one? - Mar 18th 2011, 05:24 PMalexmahone
$\displaystyle \frac{\partial f}{\partial x}=sin y$

$\displaystyle f=xsin y+A(y,z)$

$\displaystyle \frac{\partial f}{\partial y}=xcos y+z$

$\displaystyle f=xsin y+yz+B(x,z)$

$\displaystyle \frac{\partial f}{\partial z}=y$

$\displaystyle f=yz+C(x,y)$

By inspection, we get $\displaystyle f=xsin y+yz$ - Mar 19th 2011, 04:29 AMHallsofIvy
Perhaps a touch simpler: from $\displaystyle f_x= sin(y)$ we get $\displaystyle f= xsin(y)+ A(y,z)$

Now differentiate that with respect to y: $\displaystyle f_y= xcos(y)+ A_y= x cos(y)+ z$ so that $\displaystyle A_y= z$. That tells us that $\displaystyle A(y,z)= yz+ B(z)$. f(x,y,z)= x sin(y)+ A(y,z)= x sin(y)+ yz+ B(z).

Differentiating with respect to z, $\displaystyle f_z= y+ B'(z)= y$ so B'= 0 and B is a constant.

f(x,y,z)= x sin(y)+ yz+ B

Alex Mahone left off the constant because the problem only asked for "a" function having that gradient. - Mar 19th 2011, 05:42 PMxxp9
Let me add one:

let $\displaystyle \omega=f_xdx+f_ydy+f_zdz$. It's easy to verify that $\displaystyle d\omega=0$. So F is really conservative and the line integral of $\displaystyle \omega$ doesn't depend on the choice of path.

For any point $\displaystyle p=(x_0, y_0, z_0)$, Choose a path C with the following 3 segments:

I: from (0,0,0) to $\displaystyle (0, y_0, 0)$, along the y-axis

II: from $\displaystyle (0,y_0, 0)$ to $\displaystyle (0, y_0, z_0)$, along the direction of z-axis

III: from $\displaystyle (0, y_0, z_0)$ to $\displaystyle (x_0,y_0,z_0)$, along the direction of x-axis

The $\displaystyle f(p)=f(0)+\int_C \omega = f(0)+\int_I 0 dy + \int_{0}^{z_0} y_0 dz + \int_{0}^{x_0} sin(y_0)dx$

$\displaystyle =y_0 z_0 + sin(y_0) x_0 + f(0)$ - Mar 20th 2011, 01:51 AMheatly
Thanks everybody for your help.

- Mar 21st 2012, 10:49 PMJohnMbostonRe: Vector Field
I'm new to the forum.

Question: Can you please explain yow are you pasting in these nice italicized vector function equations?

I've fooled around with typing in Word and doing print screen paste or upload a jpeg but these don't seem to work.

Thanks. - Mar 22nd 2012, 12:21 AMalexmahoneRe: Vector Field
- Mar 22nd 2012, 01:58 AMblingRe: Vector Field
- Mar 22nd 2012, 11:50 PMJohnMbostonRe: Vector Field
Thanks for the tip. The Math Input Panel works for me in, say, MS Word but I cannot make entries into a Forum message box. If I copy and paste from Word I lose things like the right-pointing vector line symbol above capital F or the hats above i+j in a vector function. Suggestions?

(Sorry, I'm an email guy, not a whiz at online forums, but I'm learning.) - Mar 24th 2012, 01:50 AMblingRe: Vector Field
correct even i tried it but doesnt work. sorry mate if i find something ill post it