Final answer:
To find the function f(x, y, z) where ∇ f = (5z, 7y⁶, 5x), integrate each component of the vector with respect to its corresponding variable, considering functions of the other two variables as necessary, and ensuring no contradictions arise in the process.
Step-by-step explanation:
The student is asking how to find a function f(x, y, z) such that the gradient of f, denoted by ∇ f, is equal to the vector (5z, 7y6, 5x). In multivariable calculus, the gradient of a function is a vector that consists of all of the partial derivatives of that function with respect to each variable. In other words, for a function f(x, y, z), the gradient is given by (∂f/∂x, ∂f/∂y, ∂f/∂z).
To find such a function, we can integrate each component of the given vector with respect to its corresponding variable, while keeping in mind that the resulting expression may include functions dependent on the other variables. For example, integrating 5z with respect to x gives us a term 5xz, plus some function of y and z which we can denote as g(y, z). Similarly, integrating 7y6 with respect to y gives us (7/7)y7 plus some function of x and z, h(x, z). Lastly, integrating 5x with respect to z gives 5xz plus some function of x and y, j(x, y).
Thus, the potential form of function f(x, y, z) would be f(x, y, z) = 5xz + (7/7)y7 + j(x, y) for the corresponding term for x and y variables. Since the term 5xz appears twice in our integrals, we only consider it once in our function. The next step is to ensure that through this process, no contradictions arise between the assumed functions g(y, z), h(x, z), and j(x, y). After accounting for any such issue, the student will arrive at a function whose gradient matches the given vector.