Question

Finding Potential Functions In Exercises 7-12, find a potential function f for the field F. 7. F=2xi+3yj+4zk 8. F=(y+z)i+(x+z)j+(x+y)k 9. F=ey+2 z (i+x j+2 x k)

Answer 1

To find the potential function for F=2xi+3yj+4zk, integrate each component with respect to its variable and ensure the curl of F is zero. A potential function f(x,y,z) = x^2 + ½y^2 + 2z^2 + C is obtained, where C is a constant.

Step-by-step explanation:

The question asks for a potential function for a given vector field F. A potential function f is a scalar function whose gradient is the given vector field, i.e., F = ∇f. In mathematical terms, this involves integrating the vector field's components and ensuring that the mixed partial derivatives are equal, which is a requirement for the existence of a potential function.

To find a potential function for F=2xi+3yj+4zk, we can integrate each component with respect to its variable, provided that the field is conservative, which is the case if the field is defined on a simply connected region and its curl is zero. Assuming these conditions are met, we can start by integrating 2x with respect to x, 3y with respect to y, and 4z with respect to z to get the potential function f(x,y,z) = x2 + ½y2 + 2z2 + C, where C is the constant of integration.

Answer 2

To find a potential function for a vector field, integrate the components of the field with respect to their variables and ensure the partial derivatives are consistent. This is achieved using the fundamental theorem of calculus for line integrals, resulting in a potential function up to an arbitrary constant.

Step-by-step explanation:

The student's question involves finding a potential function f for a given vector field F. To find the potential function of a field, we typically take the integral of the vector field components with respect to their respective variables, ensuring that the partial derivatives of the potential function are consistent with the components of the vector field.

For example, if we have a field F with components Fx, Fy, and Fz, we would partially integrate Fx with respect to x, Fy with respect to y, and Fz with respect to z to find potential function candidates. We compare the mixed second-order partial derivatives to ensure that the potential function is well-defined and consistent with F. This process is a direct application of the fundamental theorem of calculus for line integrals in three dimensions which states that if a vector field F is conservative, then it has a potential function that we can find through this integration process.

It is important to note that potential functions are generally defined up to an arbitrary constant because adding a constant does not affect the derivatives and hence keeps the physical field unchanged.