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Vector Fields

For each vector field in R4 given below, either find a function for which it is the gradient, or explain why no such function exists. Variables are in the order x, y, z, w.

a.(siny + wcosx + 1)e1 + (sinz+xcosy+1)e2+(sinw+ycosz)e3+(sinx+zcosw)e4

b.yze1+xze2+xye3+ywe4
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1 Answer

a. ∇f(x,y,z,w) = 〈sin y + w cos x + 1, sin z + x cos y + 1, sin w + y cos z, sin x + z cos w〉
 
This means the following must all be f(x,y,z,w):
 
x sin y + w sin x + x + h1(y,z,w)
 
y sin z + x sin y + y + h2(x,z,w)
 
z sin w + y sin z + h3(x,y,w)
 
w sin x + z sin w + h4(x,y,z)
 
We see that these expressions can be made equal. You get:
 
f(x,y,z,w) = x sin y + w sin x + z sin w + y sin z + x + y + C
 
b. ∇f(x,y,z,w) = 〈yz, xz, xy, yw〉
 
This means the following must all be f(x,y,z,w):
 
xyz + h1(y,z,w)
 
xyz + h2(x,z,w)
 
xyz + h3(x,y,w)
 
yw2/2 + h4(x,y,z)
 
If there was an f(x,y,z,w), then we see that yw2/2 would have to be in the definition of h2(x,z,w). Therefore it doesn't exist.