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?
$\displaystyle \frac{\partial f}{\partial x}=sin y$
$\displaystyle f=xsin y+A(y,z)$
$\displaystyle \frac{\partial f}{\partial y}=xcos y+z$
$\displaystyle f=xsin y+yz+B(x,z)$
$\displaystyle \frac{\partial f}{\partial z}=y$
$\displaystyle f=yz+C(x,y)$
By inspection, we get $\displaystyle f=xsin y+yz$
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.
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 $\displaystyle f(p)=f(0)+\int_C \omega = f(0)+\int_I 0 dy + \int_{0}^{z_0} y_0 dz + \int_{0}^{x_0} sin(y_0)dx$
$\displaystyle =y_0 z_0 + sin(y_0) x_0 + f(0)$
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.
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.)