Consider the vector field ,y, z)=(5z++(4z++(4y+

a. find a function such that =∇ and ,0,0)=0.

b. suppose c is any curve from (0,0,0) to (1,1,1). use part a. to compute the line integral ∫⋅

a)

b) 10

Step-by-step explanation:

a) Here

Since the case  =  ∇ holds, then

∇ =

So,

If we integrate with respect to x, we will get an integration constant C which is also a function that depends to y and z.

Hence,

Now we need to find g(y,z).

So first let's take the derivative of g(y,z) with respect to y.

Hence,

So now, if we integrate with respect to y to find g(y,z)

Thus,

And since , then

Thus,

b) By the Fundamental Theorem of Line Integrals, we know that

Hence,