Vector Fields

Question

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

Answer 1

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.