Question

At least one of the answers above is NOT correct. 1 of the questions remains unanswered.

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.
f(x,y,z) =____

Answer 1

The resulting potential function is f(x, y, z) = (1/2)x² + (1/2)y² + (1/2)z².

Step-by-step explanation:

The student is seeking a potential function f(x, y, z) such that the gradient of f, denoted as ∇f, is equal to the vector field F(x, y, z) = xi + yj + zk. The condition that f(0,0,0) = 0 provides a specific value for the function at the origin. To find the function f, we can integrate the components of F with respect to their corresponding variables:

• To find the x-component of f, we integrate x with respect to x, obtaining (1/2)x².
• For the y-component, we integrate y with respect to y, yielding (1/2)y².
• Lastly, integrating z with respect to z gives us (1/2)z².

Thus, the potential function that satisfies all the conditions is f(x, y, z) = (1/2)x² + (1/2)y² + (1/2)z².