Consider the vector field f(x, y, z) = xi + yj + zk. Find a function f such that f = ∇f and f(0, 0, 0) = 0.

Solution:

Vectors are geometrical entities that have magnitude and direction. 

A vector can be represented by a line with an arrow pointing towards its direction and its length represents the magnitude of the vector.

∇f = f = xi + yj + zk

δf/δx = x

f(x, y, z) = x²/2 + g(y, z)

δf/δy = y = δg/δy

g(y, z) = y²/2 + h(z)

δf/δz = z = δh/δz

h(z) = z²/2 + C

We know that

f(x, y, z) = x²/2 + y²/2 + z²/2 + C

f(0, 0, 0) = 0 

C = 0

So we get

f(x, y, z) = x²/2 + y²/2 + z²/2

Therefore, a function f is f(x, y, z) = x²/2 + y²/2 + z²/2.

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Consider the vector field f(x, y, z) = xi + yj + zk. Find a function f such that f = ∇f and f(0, 0, 0) = 0.

Summary:

Consider the vector field f(x, y, z) = xi + yj + zk. A function f such that f = ∇f and f(0, 0, 0) = 0 is f(x, y, z) = x²/2 + y²/2 + z²/2.