Finding a Scalar Function f from a Vector Function F

In multivariable calculus, we often encounter situations where we are given a vector function F and asked to find a scalar function f such that the gradient of f is equal to F. This is a useful concept in optimization and other areas of mathematics.

Let's consider the following problem:

Given the vector function F = (6xyz + 2sin(x), 3x^2z, 3x^2y), find a function f so that F = ∇f, and f(0, 0, 0) = 0.

To solve this problem, we need to follow these steps:

1. Understand the problem: We are given a vector function F and asked to find a scalar function f whose gradient is equal to F. Additionally, we are told that the value of f at the point (0, 0, 0) should be 0.

2. Compute the partial derivatives of the components of F: The vector function F is given as F = (6xyz + 2sin(x), 3x^2z, 3x^2y). To find the scalar function f, we need to compute the partial derivatives of each component of F.

• Partial derivative of the first component with respect to x: ∂/∂x (6xyz + 2sin(x)) = 6yz + 2cos(x)
• Partial derivative of the first component with respect to y: ∂/∂y (6xyz + 2sin(x)) = 6xz
• Partial derivative of the first component with respect to z: ∂/∂z (6xyz + 2sin(x)) = 6xy
• Partial derivative of the second component with respect to x: ∂/∂x (3x^2z) = 6xz
• Partial derivative of the second component with respect to y: ∂/∂y (3x^2z) = 0
• Partial derivative of the second component with respect to z: ∂/∂z (3x^2z) = 3x^2
• Partial derivative of the third component with respect to x: ∂/∂x (3x^2y) = 6xy
• Partial derivative of the third component with respect to y: ∂/∂y (3x^2y) = 3x^2
• Partial derivative of the third component with respect to z: ∂/∂z (3x^2y) = 0

1. Construct the scalar function f: To find the scalar function f, we need to integrate the partial derivatives of the components of F with respect to the corresponding variables. The integration constants will be determined by the condition f(0, 0, 0) = 0.

• Integrating the partial derivatives with respect to x: ∫ (6yz + 2cos(x)) dx = 6xyz + 2sin(x) + C1(y, z)

• Integrating the partial derivatives with respect to y: ∫ 6xz dy = 3x^2z + C2(x, z)

• Integrating the partial derivatives with respect to z: ∫ 6xy dz = 3x^2y + C3(x, y)

  Combining these integrals, we get the scalar function f: f(x, y, z) = 6xyz + 2sin(x) + 3x^2z + 3x^2y + C

1. Determine the integration constant C: To satisfy the condition f(0, 0, 0) = 0, we need to set the integration constant C = 0.

Therefore, the scalar function f that satisfies the given conditions is: f(x, y, z) = 6xyz + 2sin(x) + 3x^2z + 3x^2y

We have successfully found the scalar function f whose gradient is the given vector function F = (6xyz + 2sin(x), 3x^2z, 3x^2y), and f(0, 0, 0) = 0.

This problem demonstrates the important concept of finding a scalar function from a given vector function in multivariable calculus. Understanding this process is crucial for optimization, vector field theory, and various applications in mathematics and physics.