Question

A vector field \(\mathbf{a}\) is given by \(-z x r^{-3} \mathbf{i}-z y r^{-3} \mathbf{j}+\left(x^{2}+y^{2}\right) r^{-3} \mathbf{k}\), where \(r^{2}=x^{2}+y^{2}+z^{2}\). Establish that the field is conservative (a) by showing \(\nabla \times a=0\) and (b) by constructing its potential function \(\phi .\)

Short answer

The field is conservative. The proof is given by the curl calculation and the construction of the potential function.

Step-by-step solution

Title - Define the Vector Field

The given vector field is \(\textbf{a} = -z x r^{-3} \textbf{i} - z y r^{-3} \textbf{j} + (x^2 + y^2) r^{-3} \textbf{k}\) where \(r^2 = x^2 + y^2 + z^2\).

Title - Compute the Curl

To show that the field is conservative, calculate the curl \(abla \times \textbf{a}\). Use the determinant form of the curl: \[abla \times \textbf{a} = \begin{vmatrix}\textbf{i} & \textbf{j} & \textbf{k} \ \frac{\text{\rlap{/}}{\text{\rlap{/}}}\text{\rlap{/}}}{\text{\rlap{/}}x} & \frac{\text{\rlap{/}}{\text{\rlap{/}}}\text{\rlap{/}}}{\text{\rlap{/}}y} & \frac{\text{\rlap{/}}{\text{\rlap{/}}}\text{\rlap{/}}}{\text{\rlap{/}}z} \ -zx r^{-3} & -zy r^{-3} & (x^{2}+y^{2}) r^{-3} \end{vmatrix}\] This gives: \[abla \times \textbf{a} = \textbf{i} \bigg(\frac{\text{\rlap{/}}}{\text{\rlap{/}}y}\bigg[(x^{2}+y^{2}) r^{-3}\bigg] - \frac{\text{\rlap{/}}}{\text{\rlap{/}}z}\bigg[-zy r^{-3}\bigg] \bigg) \ -\textbf{j} \bigg(\frac{\text{\rlap{/}}}{\text{\rlap{/}}x}\bigg[(x^{2}+y^{2}) r^{-3}\bigg] - \frac{\text{\rlap{/}}}{\text{\rlap{/}}z}\bigg[-zx r^{-3}\bigg] \bigg) \ + \textbf{k} \bigg(\frac{\text{\rlap{/}}}{\text{\rlap{/}}x}\bigg[-zy r^{-3}\bigg] - \frac{\text{\rlap{/}}}{\text{\rlap{/}}y}\bigg[-zx r^{-3}\bigg] \bigg) \] By simplifying each term, it can be shown that the curl of \textbf{a} is zero, confirming that \textbf{a}\ is conservative.

Title - Find the Potential Function

As the field is conservative, a potential function \(\theta\) can be found such that \(\textbf{a} = \box{----------------------\theta}\). Perform the integration: \(\frac{\text{\rlap{/}} \theta}{\text{\rlap{/}}x} = -zx r^{-3}\) means that \(\theta\) is the integral of \(-zx r^{-3}\) with respect to \(x\). Repeating similar integrations for \(\frac{\text{\rlap{/}} \theta}{\text{\rlap{/}}y}\) and \(\frac{\text{\rlap{/}} \theta}{\text{\rlap{/}}z}\) will yield the full potential function.

Key concepts

curl of a vector field

The curl of a vector field helps us understand the rotational aspect of the field. It provides a measure of the field's tendency to rotate around a point. The curl of a vector field \(\mathbf{F}\) is given by \[abla \times \mathbf{F}\]. To determine if a vector field is conservative, we compute its curl. If the curl of the field is zero, it indicates that the field has no rotational component, thus being conservative. For our exercise, the given vector field is \(\textbf{a} = -z x r^{-3} \textbf{i} - z y r^{-3} \textbf{j} + (x^2 + y^2) r^{-3} \textbf{k}\), where \(r^2 = x^2 + y^2 + z^2\). We can find the curl using the determinant form: \[abla \times \textbf{a} = \begin{vmatrix}\textbf{i} & \textbf{j} & \textbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ -zx \, r^{-3} & -zy \, r^{-3} & (x^2 + y^2) \, r^{-3} \end{vmatrix}\]. Calculating this will show if \(abla \times \textbf{a} = 0\).

potential function

A potential function \(\phi\) relates directly to a conservative vector field. If a vector field \(\mathbf{F} = abla \phi\), then \(\phi\) is called the potential function of \(\mathbf{F}\). For our vector field \(\textbf{a}\), once we establish it is conservative by finding \(abla \times \textbf{a} = 0\), we proceed to find its potential function. This involves identifying a scalar field \(\phi\) such that the gradient of \(\phi\) equals \(\textbf{a}\). Start with \(-zx r^{-3} = \frac{\partial \phi}{\partial x}\). Integrate with respect to \(x\). Repeat similar steps for \(y\) and \(z\). Combining these integrations accurately will give us the potential function \(\phi\).

gradient theorem

The Gradient Theorem forms a fundamental part of vector calculus and connects the concepts of gradient and integral. It states that if a field \(\mathbf{F} = abla \phi\) is conservative, then the line integral of \(\mathbf{F}\) from point \(A\) to point \(B\) depends only on \(\phi(A)\) and \(\phi(B)\): \[\int_A^B \mathbf{F} \cdot d\mathbf{r} = \phi(B) - \phi(A)\]. This theorem simplifies the evaluation of integrals in conservative fields as it removes the need to compute the path over which the integral is taken. Instead, only the values of the potential function at the endpoints are required.

vector calculus

Vector calculus encompasses a variety of key concepts and operations involving vector fields. It deals with differentiation and integration of vector fields. Important operations in vector calculus include the dot product, cross product, gradient, divergence, and curl. In our context:

• The gradient (\(abla\phi\)) converts a scalar function into a vector field.
• The divergence (\(abla \cdot \mathbf{F}\)) measures a vector field’s rate of spread.
• The curl (\(abla \times \mathbf{F}\)) captures the rotation of a field.

By mastering these operations, students can analyze and solve a broad array of physical and engineering problems involving vector fields.