Question

Determine whether the vector field \(F\) is conservative. If it is, find a potential function for the vector field. $$ \mathbf{F}(x, y, z)=y^{2} z^{3} \mathbf{i}+2 x y z^{3} \mathbf{j}+3 x y^{2} z^{2} \mathbf{k} $$

Short answer

The vector field \(F = y^{2} z^{3} i + 2 x y z^{3} j + 3 x y^{2} z^{2} k\) is found to be conservative and the potential function of the vector field is \(f = x y^{2} z^{3} + c\), where \(c\) is an arbitrary constant.

Step-by-step solution

Compute the curl of the vector field

The curl of a vector field \(F = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k\) is given by \(\nabla \times F =\left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)i - \left(\frac{\partial R}{\partial x} - \frac{\partial P}{\partial z}\right)j + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)k\). So, for our vector field \(F = y^{2} z^{3} i + 2 x y z^{3} j + 3 x y^{2} z^{2} k\), we first differentiate each component with respect to the appropriate variables, then substitute into the formula for the curl.

Check if the curl is zero

After calculating the curl, we need to check if it is zero. If it is zero, then the vector field \(F\) is conservative.

Find the potential function

If \(F\) is a conservative vector field, we can find a potential function \(f\) such that \(\nabla f = F\). We integrate each component of \(F\) with respect to the corresponding variable \(x\), \(y\), and \(z\) to find \(f\).

Key concepts

Curl of a Vector Field

Understanding the curl of a vector field is crucial for identifying conservative fields, which are central to many problems in vector calculus. The curl measures the rotation of a vector field around a point. Imagine a tiny paddle wheel placed in a fluid flow; the way the flow spins the wheel is analogous to the curl at that point.

Mathematically, for a vector field \( F = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k \), the curl is defined as the cross product of the del operator \( abla \) and \( F \), given by \( abla \times F \). This results in a new vector that represents the rotational tendencies of the original vector field at each point. In detail, the components of this new vector are derived from the partial derivatives of the functions that make up \( F \), as shown in the solution steps for the exercise.

When the curl of a vector field is zero at every point, it implies that there is no rotational effect present in the field—the hallmark of a conservative field. Checking if the curl is zero is therefore a standard step in identifying conservative vector fields.

Potential Function

A potential function, often denoted as \( f \), is associated with conservative vector fields. The existence of a potential function is a defining feature of these fields. One can think of this function as a way to capture the work done by or against a field when moving along a path within it.

In terms of calculus, a potential function \( f \) for a vector field \( F \) is a scalar function whose gradient \( abla f \) is equal to the vector field: \( abla f = F \). To find this function, you'd integrate the components of \( F \) with respect to the corresponding variable, while considering the integration constants carefully, to ensure that the original field is reconstructed when taking the gradient of \( f \).

A potential function is extremely useful because it simplifies the process of calculating work along a path and can also serve to determine the potential energy associated with a force field.

Gradient Field

Another key concept linked closely with potential functions is the gradient field. A gradient field \( abla f \) results from taking the gradient of a scalar function \( f \), which gives rise to a vector field. The gradient points in the direction of the greatest rate of increase of the function, and its magnitude tells you how steep the increase is.

When a vector field \( F \) is the gradient of some scalar function \( f \), the field is considered conservative. In practical terms, this property means that if you move from one point to another in a gradient field, the work done by the corresponding force is independent of the path taken, relying only on the start and end points.

This concept is very intuitive in the context of potential energy: no matter how you move around in a conservative force field (like gravity near Earth's surface), the potential energy difference depends only on the vertical displacement, not on the path you took to get there.

Partial Derivatives

Partial derivatives are the building blocks for many operations in vector calculus, including finding the curl and the gradient of a vector field. They represent the rate at which a function changes as one of its variables changes, while all other variables are held constant.

To compute the curl or to find a potential function, you need to understand how to correctly take the partial derivative of each component function within the vector field with respect to its variables. These derivatives provide the necessary values to plug into the formulas for the curl and for the components of the gradient when searching for a conservative field and its potential function.

Understanding and calculating partial derivatives accurately is a fundamental skill in vector calculus and is essential for working with vector fields, making it a topic of utmost importance in exercises like the one provided.