Problem 215 - Calculus Volume 3

Question

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

Step-by-Step Solution

Answer

The curl of \( \mathbf{F} \) is \( (xz^2 - xy^2)\mathbf{i} - (yz^2 - x^2y)\mathbf{j} + (y^2z - x^2z)\mathbf{k} \).

Step 1: Understand the Formula for Curl

The curl of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is given by the formula: \( abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix} \). For this problem, \( P = x^2 yz, \ Q = xy^2 z, \ R = xyz^2 \).

Step 2: Set Up the Determinant

Using the formula for curl, set up the determinant: \[abla \times \mathbf{F} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \x^2 yz & xy^2 z & xyz^2 \end{vmatrix}\]

Step 3: Calculate the Determinant's First Component

The first component (for \( \mathbf{i} \)) is calculated as follows:\[ \mathbf{i} \left( \frac{\partial}{\partial y}(xyz^2) - \frac{\partial}{\partial z}(xy^2z) \right) = \mathbf{i} \left( xz^2 - xy^2 \right) \]

Step 4: Calculate the Determinant's Second Component

The second component (for \( \mathbf{j} \)) is:\[ -\mathbf{j} \left( \frac{\partial}{\partial x}(xyz^2) - \frac{\partial}{\partial z}(x^2yz) \right) = -\mathbf{j} \left( yz^2 - x^2y \right) \]

Step 5: Calculate the Determinant's Third Component

The third component (for \( \mathbf{k} \)) is:\[ \mathbf{k} \left( \frac{\partial}{\partial x}(xy^2z) - \frac{\partial}{\partial y}(x^2yz) \right) = \mathbf{k} \left( y^2z - x^2z \right) \]

Step 6: Combine the Components

Combine the components from Steps 3, 4, and 5 to find the curl:\[ abla \times \mathbf{F} = (xz^2 - xy^2)\mathbf{i} - (yz^2 - x^2y)\mathbf{j} + (y^2z - x^2z)\mathbf{k} \]

Key Concepts

Curl of a Vector Field

The curl of a vector field is a concept in multivariable calculus that measures a field's tendency to rotate around a point. Imagine you are studying a fluid that flows, and you drop a small stick into it. If the stick rotates, the fluid has what we call "curl." This concept helps us understand rotational motion in vector fields.

For a vector field represented as \( \mathbf{F}(x, y, z) = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the curl is mathematically described by the cross product: \[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix} \]

The components \(P, Q, R\) can represent physical quantities like velocity in different directions. Calculating the curl of \( \mathbf{F} \) involves setting up and calculating a determinant, which brings us to the next core concept.

Determinant Calculation

Determinants are crucial for calculating the curl of a vector field. In our context, determinants help us find the rotational aspects of a vector field using partial derivatives. The determinant is expressed as a square matrix. For curl, the determinant is always a 3x3 matrix. To solve it, break it down into manageable parts.

To calculate, set up your matrix first:

• The top row contains unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \).
• The second row has the partial derivative operators \( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \).
• The third row contains the vector field components \( P, Q, R \).

The determinant is solved by expanding across the first row (unit vectors). This process involves multiplying an element from the first row by the determinant of a 2x2 matrix formed by removing the row and column of the element, then subtracting and adding accordingly.

Determinants transform complicated calculations involving vectors into simpler arithmetic.

Partial Derivatives

Partial derivatives are a fundamental tool in the study of multivariable calculus. They show us how a function changes as just one of its variables change while the others remain constant. This insight is invaluable when analyzing vector fields and other multivariable functions.

Consider a scenario where you have a function depending on multiple variables like \( P(x, y, z) = x^2 yz \). The partial derivative with respect to \( x \) would be found by treating \( y \) and \( z \) as constants, resulting in \( \frac{\partial P}{\partial x} = 2xyz \). Similarly, we find derivatives with respect to \( y \) and \( z \) by keeping the other variables constant.

In the context of curl, partial derivatives tell us how tiny changes in the variables affect the vector field's components, thus allowing us to determine incrementally how the field "twists" or "rotates." This makes partial derivatives a powerful analytic tool to assess intricate relationships in mathematical models. They are essentially the building blocks for understanding directional change in multivariable functions.