(b) Find a potential f for F. That is, find a function f such that F = Vƒ (Hint: In class we integrated with respect to each variable and ended up with certain unknown functions of the other variable. This problem does not need to be this complicated, remember that any potential function will do.)

The problem is to find a function f such that F=∇f=fx,fy,fz Find f such that fx=excos y, fy=-exsiny, fz=2…1 In order to find f integrate fx with respect to x. ∫fxdx=∫excos y dxf=cosy∫ex dxf=excosy+c Here since fx was integrated with respect to x, the constant c must be a function of y and z. Therefore, f=excosy+hy,z To find hy,z differentiate the function f found in the previous step with respect to z. fz=∂hy,z∂z From 1, fz must be equal to 2. Therefore, ∂hy,z∂z=2. Now differentiate the function f found in the previous step with respect to y. Then, fy=-exsiny+∂hy,z∂y From 1, fy must be equal to -exsiny Therefore, it follows that ∂hy,z∂y=0 This implies that h is not dependent on y and hence h is a function of z alone. Therefore, ∂hy,z∂z=dhzdz=2 This means, h'z=2……2 Integrating 2 with respect to z, we get, hz=2z+c, where c is a real constant Therefore f=excosy+2z+c, where c is a real constant The potential f for F is, f=excosy+2z+c, where c is a real constant

Answered: (b) Find a potential f for F. That is, find a function f such that F = Vƒ (Hint: In class we integrated with respect to each variable and ended up with certain…

Advanced Engineering Mathematics

10th Edition

ISBN: 9780470458365

Author: Erwin Kreyszig

Publisher: Wiley, John & Sons, Incorporated

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A vector field with component form F(x, y, 2) = (P(x, y, z), Q(x, y, 2), R(x, y, 2)) on
R³ is conservative if
ӘР ӘР
de
dy' dz
ƏR
and
dz
ƏR
ду
provided that certain "nice" conditions are met (which we assume).
Let F(x, y, z) = (e* cos y, -eª sin y, 2).
(b) Find a potential f for F. That is, find a function f such that F = Vf (Hint:
In class we integrated with respect to each variable and ended up with certain
unknown functions of the other variable. This problem does not need to be this
complicated, remember that any potential function will do.)

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