Question

Determine whether the vector field \(F\) is conservative. If it is, find a potential function for the vector field. $$ \mathbf{F}(x, y, z)=e^{z}(y \mathbf{i}+x \mathbf{j}+\mathbf{k}) $$

Short answer

To determine whether the vector field \(F\) is conservative, calculate the curl of \(F\), and check if it is equal to zero. If \(F\) is conservative, the potential function can be found by solving the partial differential equations \(\nabla f = F\). The final answer will be a function of \(x, y, z\).

Step-by-step solution

Compute the curl of \(F\)

To start with, calculate the curl of the vector field \(F\), which is a determinant involving the unit vectors \(i, j, k\), the gradient \(\nabla\) and the vector field \(F\). In this case, \(\nabla \times F = \nabla \times (e^{z}(y \mathbf{i}+x \mathbf{j}+\mathbf{k}))\). After calculation, if the curl of \(F\) is zero, then the field \(F\) is conservative.

Determine whether the curl is equal to zero

Determine if the curl found in Step 1 equals to zero. If \(\nabla \times F = 0\), then \(F\) is conservative, otherwise, \(F\) is not conservative.

Find the potential function

If \(F\) is conservative, by integration, find the function \(f(x, y, z)\) such that \(F = \nabla f = (f_x, f_y, f_z)\). Here, \(f_x, f_y, f_z\) represent the partial derivatives of \(f\) with respect to \(x, y, z\) respectively. In this case, expect to solve: \(f_x = y e^z, f_y = x e^z, f_z = e^z\)

Solve for \(f\)

Finally, integrate the partial differential equations in Step 3 to find the function \(f\). The final function \(f(x, y, z)\) must satisfy all three equations simultaneously, and might have an arbitrary constant.

Key concepts

Potential Function

Understanding the concept of a potential function is foundational when studying conservative vector fields. In physics, a conservative vector field represents a force field where no energy is lost when moving a particle from one point to another. Mathematically, if a vector field F is conservative, there exists a scalar function f(x, y, z), called a potential function, such that F can be represented as the gradient of f. The gradient, indicated by \( abla f \), is a vector of the partial derivatives of the function f with respect to each variable.

When you find a potential function for a vector field, you're essentially reversing the gradient operation. To verify the existence of a potential function, it's common to first check whether the vector field is conservative, which can often be done by evaluating the curl of the vector field. If the curl is zero, then the field has a potential function. This function is useful because it simplifies the computation of work done by the force along a path and is key to many physical laws.

Curl of a Vector Field

The curl of a vector field helps us understand the rotational behavior of the field at a certain point. Formally, the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. Technically speaking, if F is the vector field in question, then the curl of F, denoted as \( abla \times F \), can be found by calculating the determinant that includes the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \), the del operator \( abla \), and the vector field components. If the resulting vector has all zero components, it means that the field is irrotational at every point; therefore, it's a sine qua non condition for the vector field to be conservative and for a potential function to exist.

For the students having trouble with this concept, remember: the curl is a measure of the 'twist' of a vector field, and when it equals zero, it's akin to a water flow with no eddies or whirlpools – perfectly straightforward and free from turbulence, embodying a conservative nature.

Partial Derivatives

Partial derivatives are a staple in multivariable calculus and are vital when dealing with functions of several variables. They are used to measure how a function changes as only one of the variables changes while the others are held constant. In the context of potential functions, if f is the scalar potential, then the partial derivatives of f, indicated as \( f_x, f_y, f_z \), represent how the function f changes in response to small changes in x, y, and z, respectively. These partial derivatives collectively make up the gradient \( abla f \), which then gives us the field F.

To find a potential function for a conservative vector field, one must often solve a system of partial differential equations obtained from equating the components of \( F \) to the respective partial derivatives of f. Solving these equations requires integration, which may introduce an arbitrary constant, indicative of the fact that potential functions can be determined up to an additive constant since gradients are not affected by constant terms.