Consider the vector field F(x, y, z) = x i + y j + z k. Find a function f such that f = F and f(0, 0, 0) = 0.

To find a function f that corresponds to the vector field F(x, y, z) = x i + y j + z k, we need to recognize that this vector field is essentially a gradient field. We seek a scalar function f(x, y, z) whose gradient yields the vector field F.

The relationship we are looking for is given by:

F = ∇f

This means that the components of the vector field F correspond to the partial derivatives of the function f:

• ∂f/∂x = x
• ∂f/∂y = y
• ∂f/∂z = z

Next, we integrate each component to find f:

1. Integrating ∂f/∂x = x gives:

f(x, y, z) = (1/2)x² + g(y, z)

2. Integrating ∂f/∂y = y gives:

f(x, y, z) = (1/2)x² + (1/2)y² + h(z)

3. Integrating ∂f/∂z = z gives:

f(x, y, z) = (1/2)x² + (1/2)y² + (1/2)z² + C

Consequently, we can deduce that the function:

f(x, y, z) = (1/2)(x² + y² + z²) + C

is suitable. Now, for the condition f(0, 0, 0) = 0, we evaluate:

f(0, 0, 0) = (1/2)(0² + 0² + 0²) + C = C

This implies that C = 0. Thus, the required function is:

f(x, y, z) = (1/2)(x² + y² + z²)

In conclusion, the function f such that F = ∇f and f(0, 0, 0) = 0 is:

f(x, y, z) = (1/2)(x² + y² + z²)