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Magazine Chapter 5: Problem 66
Find the divergence and curl for the following vector fields.\(\mathbf{F}(x, y, z)=e^{x} \mathbf{i}+e^{x y} \mathbf{j}+e^{\mathbf{x y z}} \mathbf{k}\)
Short Answer
Expert verified
Divergence: \(e^x + xe^{xy} + xy e^{xyz}\); Curl: \((xz - xy)e^{xyz} \mathbf{i} - yz e^{xyz} \mathbf{j} + y e^{xy} \mathbf{k}\)."
Step by step solution
01
Understand the Divergence Formula
The divergence of a vector field \(\mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\) is given by \(abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\). We need to apply this formula to the given vector field \(\mathbf{F}(x, y, z) = e^x \mathbf{i} + e^{xy} \mathbf{j} + e^{xyz} \mathbf{k}\).
02
Compute the Divergence
For \(P = e^x\), \(Q = e^{xy}\), and \(R = e^{xyz}\), compute the partial derivatives: \(\frac{\partial P}{\partial x} = e^x\), \(\frac{\partial Q}{\partial y} = xe^{xy}\), and \(\frac{\partial R}{\partial z} = xy e^{xyz}\). The divergence is then \(abla \cdot \mathbf{F} = e^x + xe^{xy} + xy e^{xyz}\).
03
Understand the Curl Formula
The curl of a vector field \(\mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\) is \(abla \times \mathbf{F} = (\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}) \mathbf{i} + (\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}) \mathbf{j} + (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) \mathbf{k}\).
04
Compute the Curl
Using the given \( P, Q, R \): \(\frac{\partial R}{\partial y} = xz e^{xyz}\), \(\frac{\partial Q}{\partial z} = xy e^{xyz}\), \(\frac{\partial P}{\partial z} = 0\), \(\frac{\partial R}{\partial x} = yz e^{xyz}\), \(\frac{\partial Q}{\partial x} = ye^{xy}\), and \(\frac{\partial P}{\partial y} = 0\). Thus, the curl is \(abla \times \mathbf{F} = (xz e^{xyz} - xy e^{xyz}) \mathbf{i} + (0 - yz e^{xyz}) \mathbf{j} + (y e^{xy} - 0) \mathbf{k}\).
05
Simplify the Curl
Simplify the components of the curl: \((xz - xy)e^{xyz} \mathbf{i} - yz e^{xyz} \mathbf{j} + y e^{xy} \mathbf{k}\). Therefore, the curl is \((xz - xy)e^{xyz} \mathbf{i} - yz e^{xyz} \mathbf{j} + y e^{xy} \mathbf{k}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Divergence
Divergence is a fundamental concept in vector calculus that measures how much a vector field spreads out or converges at a given point.
It is akin to understanding how much a fluid is expanding or compressing in a space. In mathematical terms, for a 3D vector field, it involves partial derivatives of each component:
It is akin to understanding how much a fluid is expanding or compressing in a space. In mathematical terms, for a 3D vector field, it involves partial derivatives of each component:
- The formula for divergence is \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \), where \(\mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\).
- For any vector field, calculate the divergence to determine if the vector field represents a source or a sink.
- Compute the partial derivatives: \( \frac{\partial P}{\partial x} = e^x \), \( \frac{\partial Q}{\partial y} = xe^{xy} \), \( \frac{\partial R}{\partial z} = xy e^{xyz} \).
- Combine them to obtain the divergence: \( abla \cdot \mathbf{F} = e^x + xe^{xy} + xy e^{xyz} \).
Curl
Curl, another key concept in vector calculus, indicates how much a vector field rotates around a point.
If you think about how water swirls in a whirlpool, that's a good analogy to curl. It provides information about the rotation and circulation at each point.
If you think about how water swirls in a whirlpool, that's a good analogy to curl. It provides information about the rotation and circulation at each point.
- The mathematical expression for curl in a 3-dimensional space is given by:\( abla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right) \mathbf{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right) \mathbf{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \mathbf{k} \).
- Curl helps explain phenomena in physics where rotation is important, like electromagnetism.
- Calculate the required partial derivatives: \( \frac{\partial R}{\partial y} = xz e^{xyz} \), \( \frac{\partial Q}{\partial z} = xy e^{xyz} \), \( \frac{\partial P}{\partial z} = 0 \), \( \frac{\partial R}{\partial x} = yz e^{xyz} \), \( \frac{\partial Q}{\partial x} = ye^{xy} \), \( \frac{\partial P}{\partial y} = 0 \).
- Combine these to find the curl: \( abla \times \mathbf{F} = (xz - xy)e^{xyz} \mathbf{i} - yz e^{xyz} \mathbf{j} + y e^{xy} \mathbf{k} \).
Partial Derivatives
Partial derivatives are the backbone of both divergence and curl because they allow us to measure the rate of change of a function with respect to one variable while keeping the others constant.
They are crucial in understanding multivariable functions, like those found in vector calculus.
They are crucial in understanding multivariable functions, like those found in vector calculus.
- Each component of a vector field can be treated separately when taking partial derivatives.
- This means examining how the vector changes in the direction of the unit vectors \( \mathbf{i}, \mathbf{j} \text{ and } \mathbf{k} \).
- The partial derivative \( \frac{\partial P}{\partial x} = e^x \) examines change along x-direction for component \( P \).
- Similarly, \( \frac{\partial Q}{\partial y} = xe^{xy} \) examines change along y-direction for component \( Q \).
- And \( \frac{\partial R}{\partial z} = xy e^{xyz} \) captures change along z-direction for component \( R \).
Vector Fields
Vector fields are visual representations of a function that assigns a vector to every point in space.
Imagine a map where each location has an arrow pointing in some direction; that's a vector field.
Imagine a map where each location has an arrow pointing in some direction; that's a vector field.
- These fields are used to model various phenomena in physics, like wind patterns, electromagnetic fields, or fluid flows.
- A vector field is described in terms of its components, often denoted by \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \).
- The component \( e^x \mathbf{i} \) might represent an exponential increase in the x-direction.
- The component \( e^{xy} \mathbf{j} \) indicates interactions between x and y directions.
- The component \( e^{xyz} \mathbf{k} \) implies a complex relation involving x, y, and z directions.
