Question

Find the curl of the vector field \(F\). \(\mathbf{F}(x, y, z)=e^{x^{2}+y^{2}} \mathbf{i}+e^{y^{2}+z^{2}} \mathbf{j}+x y z \mathbf{k}\)

Short answer

\< \text{{The curl of the given vector field}} \mathbf{F} \text{{is}} \: (2ze^{y^2 + z^2} - xz) \mathbf{i} - yz \mathbf{j} + 2xe^{x^2 + y^2} \mathbf{k} \>

Step-by-step solution

Set up the Matrix

Write the curl as a matrix and set it up as follows: \[\text{{curl}} F = \text{{det}} \begin{bmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ e^{x^2 + y^2} & e^{y^2 + z^2} & xyz \\end{bmatrix}\]

Expand the Determinant

To find the determinant of the 3x3 matrix, expand along the first row:\[ \begin{align*}\mathbf{i} \cdot \left(\frac{\partial}{\partial y}(xyz) - \frac{\partial}{\partial z}(e^{y^2 + z^2})\right) - \mathbf{j} \cdot \left(\frac{\partial}{\partial x}(xyz) - \frac{\partial}{\partial z}(e^{x^2 + y^2})\right) + \mathbf{k} \cdot \left(\frac{\partial}{\partial x}(e^{y^2 + z^2}) - \frac{\partial}{\partial y}(e^{x^2 + y^2})\right)\end{align*}\]

Evaluate the Partial Derivatives

Now compute each partial derivative: \(\frac{\partial}{\partial y}(xyz) = xz, \quad \frac{\partial}{\partial z}(e^{y^2 + z^2}) = 2ze^{y^2 + z^2} \)\(\frac{\partial}{\partial x}(xyz) = yz, \quad \frac{\partial}{\partial z}(e^{x^2 + y^2}) = 0\)\(\frac{\partial}{\partial x}(e^{y^2 + z^2}) = 0, \quad \frac{\partial}{\partial y}(e^{x^2 + y^2}) = 2xe^{x^2 + y^2} \)Substitute all the computed values back in to get the curl.

Substitute the Values

After substituting the computed values, the final curl of F is given as follows:\[\text{{curl}} F = (2ze^{y^2 + z^2} - xz) \mathbf{i} - yz \mathbf{j} + 2xe^{x^2 + y^2} \mathbf{k}\]

Key concepts

Partial Derivatives

Understanding partial derivatives is essential when studying vector calculus and analyzing the behavior of multivariable functions, like vector fields. Think of partial derivatives as a way to measure how a function changes along one axis while keeping the others constant.

Let's look at a vector field \( F(x, y, z) \) that depends on three variables. If we want to understand how this field changes only in response to \( x \) while \( y \) and \( z \) remain fixed, we compute the partial derivative with respect to \( x \) denoted as \( \frac{\partial}{\partial x} F \). With our example \(\mathbf{F}(x, y, z)=e^{x^{2}+y^{2}} \mathbf{i}+e^{y^{2}+z^{2}} \mathbf{j}+x y z \mathbf{k}\), the partial derivative with respect to \( x \) for each component of the field gives us a piece of the puzzle to find the curl.

A typical mistake when computing partial derivatives is to not correctly hold the other variables constant, but careful step-by-step calculation helps avoid this pitfall. For instance, the partial derivative of \( xyz \) with respect to \( y \) is \( xz \), considering only the rate of change in the \( y \) direction.

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields and operations applied to these fields, such as divergence, gradient, and curl. The curl operation, in particular, measures the rotation of a vector field around a point.

In the given exercise, the goal is to find the curl of the vector field \( \mathbf{F} \). A handy mnemonic for remembering how to set up the curl is the 'del operator crossed with the vector field.' You can visualize the action of the curl as the circulation, or whirling, of the vector field at a point, much like how water flows around a drain.

To improve the understanding of the curl, imagine a tiny paddle wheel placed within the vector field. If the field induces a spin on this wheel, the curl at that point is non-zero, signifying rotation in the field. The direction of this rotation corresponds to the 'right-hand rule,' which aligns with the curl's vector direction.

Matrix Determinant

The determinant of a matrix is a special number that provides valuable properties and information about the matrix, such as whether its inverse exists and its volume distortion factor in linear transformations. When dealing with the curl of a vector field, we use a 3x3 matrix whose determinant helps us find the resulting vector.

In our problem, we begin by creating a 3x3 matrix with the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) and the partial derivatives of the vector field components. The calculation of the determinant is performed by expanding along a row or column, which in this context, is often the top row where the unit vectors are. A common error students might make is forgetting the signs associated with each minor when they expand the matrix determinant. It is crucial to alternate signs starting with a positive for the top-left term.

The computation of a matrix determinant, especially for 3x3 matrices, can seem daunting. Still, by breaking the process down step by step and carefully evaluating each minor, the task becomes manageable and less error-prone, leading to the accurate calculation of the curl.