Perhaps a touch simpler: from we get
Now differentiate that with respect to y: so that . That tells us that . f(x,y,z)= x sin(y)+ A(y,z)= x sin(y)+ yz+ B(z).
Differentiating with respect to z, 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 . It's easy to verify that . So F is really conservative and the line integral of doesn't depend on the choice of path.
For any point , Choose a path C with the following 3 segments:
I: from (0,0,0) to , along the y-axis
II: from to , along the direction of z-axis
III: from to , along the direction of x-axis
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 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.)