Problem 9 - Calculus: Early Transcendental Functions

Question

Find the curl and divergence of the given vector field. $$\left\langle 3 y z, x^{2}, x \cos y\right\rangle$$

Step-by-Step Solution

Answer

The curl of the vector field \(\langle 3 y z, x^{2}, x \cos y\rangle\) is \(\langle 0,0, 2x - 3z \rangle \) and the divergence is \( 2x + 3z - x \sin y \)

Step 1: Calculation of Curl

The curl of a vector field is found using the formula: \n\n\[\n\text{Curl}(F) = \nabla \times F = ( \frac{∂f3}{∂y} - \frac{∂f2}{∂z}, \frac{∂f1}{∂z} - \frac{∂f3}{∂x}, \frac{∂f2}{∂x} - \frac{∂f1}{∂y})\n\]\n\nHere, \(\nabla \times F\) represents the curl, \(F1, F2, F3\) represent the components of the given vector field and the symbol \(\partial/ \partial x\) represents partial differentiation with respect to \(x\), \(\partial/ \partial y\) represents partial differentiation with respect to \(y\) and \(\partial/ \partial z\) represent partial differentiation with respect to \(z\). Let's plug in the components of the given vector field to calculate the curl.

Step 2: Results for Curl

Performing the calculation in step 1, the curl of the vector field \(\langle 3 y z, x^{2}, x \cos y\rangle\) is found to be \(\langle 0,0, 2x - 3z \rangle \)

Step 3: Calculation of Divergence

The divergence of a vector field is found using the formula: \n\n\[\n\text{Div}(F) = \nabla \cdot F = \frac{∂f1}{∂x} + \frac{∂f2}{∂y} + \frac{∂f3}{∂z}\n\]\n\nHere, \(\nabla \cdot F\) represents the divergence, \(F1, F2, F3\) represent the components of the given vector field and the symbol \(\partial/ \partial x\) represents partial differentiation with respect to \(x\), \(\partial/ \partial y\) represents partial differentiation with respect to \(y\) and \(\partial/ \partial z\) represent partial differentiation with respect to \(z\). Now we plug in the components of the given vector field to calculate the divergence.

Step 4: Results for Divergence

Performing the calculation in step 3, the divergence of the vector field \(\langle 3 y z, x^{2}, x \cos y\rangle\) is found to be \( 2x + 3z - x \sin y \)

Key Concepts

Curl of a Vector Field

When you think about the *curl* of a vector field, imagine it describes how the field rotates around a point. A useful analogy is the hidden spin or whirlpool-like effect in a fluid flow. In math terms, the curl gives us a new vector that reveals rotational properties of the original field.

To calculate curl, we use a special operation called the cross product. This cross product involves partial derivatives, each focusing on one variable at a time. For a vector field \( \langle f_1, f_2, f_3 \rangle \), the curl is computed as:

• \( \text{Curl}(F) = abla \times F = \left(\frac{\partial f_3}{\partial y} - \frac{\partial f_2}{\partial z}, \frac{\partial f_1}{\partial z} - \frac{\partial f_3}{\partial x}, \frac{\partial f_2}{\partial x} - \frac{\partial f_1}{\partial y}\right) \)

For the vector field \(\langle 3 y z, x^{2}, x \cos y \rangle \), finding the curl gives us the vector \(\langle 0, 0, 2x - 3z \rangle \), indicating that the major rotational component is along the z-direction.

Divergence of a Vector Field

The *divergence* of a vector field reveals how much a field expands or contracts at a point. Think of it as measuring how a cluster of particles within the field is growing, shrinking, or remaining constant.

To find the divergence, we use the dot product of a differential operator \( abla \) with the vector field itself. Mathematically, the divergence of the vector field \( \langle f_1, f_2, f_3 \rangle \) is given by:

• \( \text{Div}(F) = abla \cdot F = \frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y} + \frac{\partial f_3}{\partial z} \)

For example, in the field \(\langle 3 y z, x^{2}, x \cos y \rangle\), the formula results in a divergence of \(2x + 3z - x \sin y\). This result tells us about areas where the field is spreading out or pulling together.

Partial Differentiation

*Partial differentiation* allows us to focus on the influence of a single variable while keeping others fixed, which is essential in multivariable calculus. This comes in handy when dealing with vector fields as they depend on several variables.

Consider a function of several variables, like \( f(x, y, z) \). Partial derivatives are represented by symbols like \( \frac{\partial}{\partial x} \) which signifies differentiating with respect to \( x \) while treating \( y \) and \( z \) as constants.

• Partial derivative with respect to \( x \): \( \frac{\partial f}{\partial x} \)
• Partial derivative with respect to \( y \): \( \frac{\partial f}{\partial y} \)
• Partial derivative with respect to \( z \): \( \frac{\partial f}{\partial z} \)

In the task of finding curl and divergence of a vector field like \(\langle 3 y z, x^{2}, x \cos y \rangle\), partial differentiation is used for each component individually, allowing us to see how the field changes in each specific direction.