Problem 22 - Calculus Early Transcendentals

Question

Find div F and curl F. $$ \mathbf{F}(x, y, z)=\ln x \mathbf{i}+e^{x y z} \mathbf{j}+\tan ^{-1}(z / x) \mathbf{k} $$

Step-by-Step Solution

Answer

\( \text{div} \mathbf{F} = \frac{1}{x} + x z e^{x y z} + \frac{x}{x^2 + z^2} \); \( \nabla \times \mathbf{F} = -x y e^{x y z} \mathbf{i} + \frac{z}{x^2 + z^2} \mathbf{j} + y z e^{x y z} \mathbf{k} \).

Step 1: Understand the Divergence

The divergence of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is given by \( \text{div} \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \).

Step 2: Compute Partial Derivatives for Divergence

Given \( \mathbf{F}(x, y, z)=\ln x \mathbf{i}+e^{x y z} \mathbf{j}+\tan ^{-1}(z / x) \mathbf{k} \): - \( P = \ln x \), so \( \frac{\partial P}{\partial x} = \frac{1}{x} \).- \( Q = e^{x y z} \), so \( \frac{\partial Q}{\partial y} = x z e^{x y z} \).- \( R = \tan^{-1}(z/x) \), so \( \frac{\partial R}{\partial z} = \frac{x}{x^2 + z^2} \).

Step 3: Calculate the Divergence

Now substitute the partial derivatives into the divergence formula: \[ \text{div} \mathbf{F} = \frac{1}{x} + x z e^{x y z} + \frac{x}{x^2 + z^2} \].

Step 4: Understand the Curl

The curl of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is \[ abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} - \left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \].

Step 5: Compute Partial Derivatives for Curl

- \( \frac{\partial R}{\partial y} = 0 \) since \( \tan^{-1}(z/x) \) is independent of \( y \).- \( \frac{\partial Q}{\partial z} = x y e^{x y z} \).- \( \frac{\partial R}{\partial x} = \frac{-z}{x^2 + z^2} \).- \( \frac{\partial P}{\partial z} = 0 \) since \( \ln x \) is independent of \( z \).- \( \frac{\partial Q}{\partial x} = y z e^{x y z} \).- \( \frac{\partial P}{\partial y} = 0 \) since \( \ln x \) is independent of \( y \).

Step 6: Calculate the Curl

Now substitute the partial derivatives into the curl formula:\[ abla \times \mathbf{F} = (0 - x y e^{x y z}) \mathbf{i} - \left(\frac{-z}{x^2 + z^2} - 0\right) \mathbf{j} + (y z e^{x y z} - 0) \mathbf{k} \].Thus, \( abla \times \mathbf{F} = -x y e^{x y z} \mathbf{i} + \frac{z}{x^2 + z^2} \mathbf{j} + y z e^{x y z} \mathbf{k} \).

Key Concepts

Divergence

In vector calculus, divergence is a measure of a vector field's tendency to originate from or converge into certain points. Imagine a field of arrows, where each arrow has a direction and length; the divergence tells us if more arrows are diverging out from or converging into a point in space.
While calculating the divergence for a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the formula used is:

• \( \text{div} \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \).

This means you take the partial derivative of each component of the vector field concerning its corresponding variable.
In our exercise, given the vector \( \mathbf{F}(x, y, z)=\ln x \mathbf{i}+e^{xyz} \mathbf{j}+\tan^{-1}(z / x) \mathbf{k} \), we compute:

• For \( P = \ln x \), \( \frac{\partial P}{\partial x} = \frac{1}{x} \)
• For \( Q = e^{xyz} \), \( \frac{\partial Q}{\partial y} = xz e^{xyz} \)
• For \( R = \tan^{-1}(z/x) \), \( \frac{\partial R}{\partial z} = \frac{x}{x^2 + z^2} \)

Thus, the calculated divergence is: \( \text{div} \mathbf{F} = \frac{1}{x} + xz e^{xyz} + \frac{x}{x^2 + z^2} \).
Divergence helps determine sources or sinks within a vector field and is a crucial concept in fields like fluid dynamics and electromagnetism.

Curl

Curl, in vector calculus, measures the rotation or the "twirliness" of a vector field around a point. This concept is akin to how the water in a whirlpool swirls around.
To find the curl of a vector field, the following formula is used:

• \( \mathbf{abla} \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} - \left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \).

In our example, the vector field \( \mathbf{F}(x, y, z)=\ln x \mathbf{i}+e^{xyz} \mathbf{j}+\tan^{-1}(z / x) \mathbf{k} \), requires us to calculate these partial derivatives:

• \( \frac{\partial R}{\partial y} = 0 \)
• \( \frac{\partial Q}{\partial z} = x y e^{xyz} \)
• \( \frac{\partial R}{\partial x} = \frac{-z}{x^2 + z^2} \)
• \( \frac{\partial P}{\partial z} = 0 \)
• \( \frac{\partial Q}{\partial x} = y z e^{xyz} \)
• \( \frac{\partial P}{\partial y} = 0 \)

Substituting into the curl formula, we find:

• \( \mathbf{abla} \times \mathbf{F} = -x y e^{xyz} \mathbf{i} + \frac{z}{x^2 + z^2} \mathbf{j} + y z e^{xyz} \mathbf{k} \).

Curl plays a significant role in electromagnetism and fluid dynamics, for instance, showing how magnetic fields and fluid vortices behave.

Vector Fields

A vector field is a way of assigning vectors to each point in space. Imagine arrows that have different directions and magnitudes at various points in space, representing a vector field.
In mathematics and physics, vector fields are used to model various phenomena such as magnetic fields, velocity fields, or gravitational fields.
When examining vector fields, two critical measurements are often taken: divergence and curl, as these describe how the field behaves locally.
Consider the vector field \( \mathbf{F}(x, y, z)=\ln x \mathbf{i}+e^{xyz} \mathbf{j}+\tan^{-1}(z / x) \mathbf{k} \):

• Each component represents the vector field's behavior in the \( x \), \( y \), and \( z \) directions.
• Divergence of this field indicates how the field acts as a source or sink.
• Curl of this field gives insight into its rotation or swirling pattern.

Understanding these characteristics helps in visualizing vectors as part of larger systems, such as the forces in a fluid or air flow over a wing.
While vector fields can be complex, breaking them down through divergence and curl can significantly simplify and clarify their behaviors.