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

b.yze1+xze2+xye3+ywe4

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

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

b.yze1+xze2+xye3+ywe4

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This means the following must all be f(x,y,z,w):

x sin y + w sin x + x + h_{1}(y,z,w)

y sin z + x sin y + y + h_{2}(x,z,w)

z sin w + y sin z + h_{3}(x,y,w)

w sin x + z sin w + h_{4}(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

This means the following must all be f(x,y,z,w):

xyz + h_{1}(y,z,w)

xyz + h_{2}(x,z,w)

xyz + h_{3}(x,y,w)

yw^{2}/2 + h_{4}(x,y,z)

If there was an f(x,y,z,w), then we see that yw^{2}/2 would have to be in the definition of h_{2}(x,z,w). Therefore it doesn't exist.

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