Problem 1 Find curl $\mathbf{v}$ for \(\... [FREE SOLUTION]

Chapter 9: Problem 1

Find curl $\mathbf{v}$ for $\mathbf{v}$ given with respect to right-handed Cartesian coordinates. Show the details of your work. $$\left[\begin{array}{lll}y, & 2 x^{2}, & 0\end{array}\right]$$

Short Answer

Expert verified

The curl of the vector field is $ \langle 0, 0, 4x - 1 \rangle $.

Step by step solution

Understand the Curl Formula

To find the curl of a vector field $ \mathbf{v} = \langle P, Q, R \rangle $, we use the formula: $ abla \times \mathbf{v} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) $.

Identify Components of $\mathbf{v}$

From the given vector field $ \mathbf{v} = \langle y, 2x^2, 0 \rangle $, identify the components: $ P = y $, $ Q = 2x^2 $, and $ R = 0 $.

Compute $ \frac{\partial R}{\partial y} $ and $ \frac{\partial Q}{\partial z} $

Calculate $ \frac{\partial R}{\partial y} = \frac{\partial}{\partial y}(0) = 0 $ and $ \frac{\partial Q}{\partial z} = \frac{\partial}{\partial z}(2x^2) = 0 $.

Compute $ \frac{\partial P}{\partial z} $ and $ \frac{\partial R}{\partial x} $

Calculate $ \frac{\partial P}{\partial z} = \frac{\partial}{\partial z}(y) = 0 $ and $ \frac{\partial R}{\partial x} = \frac{\partial}{\partial x}(0) = 0 $.

Compute $ \frac{\partial Q}{\partial x} $ and $ \frac{\partial P}{\partial y} $

Calculate $ \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(2x^2) = 4x $ and $ \frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(y) = 1 $.

Apply the Curl Formula

Substitute the computed partial derivatives into the curl formula: $ abla \times \mathbf{v} = \left( 0 - 0, 0 - 0, 4x - 1 \right) $.

Write the Final Answer

The curl of the vector field is $ abla \times \mathbf{v} = \langle 0, 0, 4x - 1 \rangle $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Curl of a Vector Field

The curl of a vector field is a fascinating concept in vector calculus, representing how a vector field circulates around a point. To conceptualize this, imagine water swirling around a tiny paddle wheel. The curl measures the wheel's turning speed and direction at a point in the field.

The mathematical expression for curl applies to vector fields in 3D space.
For a vector field composed of components $ \mathbf{v} = \langle P, Q, R \rangle $, the curl is $ abla \times \mathbf{v} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) $.
The result of the curl operation is another vector, indicating how the field rotates about a point.

Understanding the details of computing the curl is important for analyzing fluid dynamics, electromagnetism, and more. For example, in this exercise, the result $ abla \times \mathbf{v} = \langle 0, 0, 4x - 1 \rangle $ shows that the rotation of the field is concentrated in the $z$-direction.

Partial Derivatives

Partial derivatives play a key role in calculating the curl of a vector field. They provide a way of understanding how a function changes with respect to one variable while keeping the others constant.

When finding the curl, partial derivatives, such as $ \frac{\partial R}{\partial y} $ or $ \frac{\partial Q}{\partial x} $, isolate the effect of changing one variable.
The symbol $ \frac{\partial }{\partial x} $ represents the rate of change with respect to $ x $.
In the given vector field $ \mathbf{v} = \langle y, 2x^2, 0 \rangle $, calculating these derivatives, such as $ \frac{\partial Q}{\partial x} = 4x $ and $ \frac{\partial P}{\partial y} = 1 $, helps assemble the curl vector.

Taking partial derivatives is akin to slicing through a multivariable function, examining its behavior from different perspectives. Mastery of this concept enhances our ability to delve into complex vector fields.

Cartesian Coordinates

Cartesian coordinates form the backbone of expressing vector fields in a familiar 3D space. It is a simple and practical system for describing locations and transformations within a vector calculus framework.

The coordinates $ (x, y, z) $ describe the position of points in 3-dimensional space.
Vector fields are often represented as $ \mathbf{v} = \langle P, Q, R \rangle $, where each component aligns with a Cartesian axis.
Using Cartesian coordinates allows us to easily apply operations like curl, gradient, and divergence on vector fields.

Thanks to the intuitive nature of Cartesian coordinates, interpreting calculations and results becomes more straightforward. In our exercise, the vector field $ \mathbf{v} = \langle y, 2x^2, 0 \rangle $ is neatly broken down into components along the $x, y, z$ axes, simplifying the process of finding the curl and illustrating the practical utility of this coordinate system.

91影视

Find curl \(\mathbf{v}\) for \(\mathbf{v}\) given with respect to right-handed Cartesian coordinates. Show the details of your work. $$\left[\begin{array}{lll}y, & 2 x^{2}, & 0\end{array}\right]$$

Short Answer

Step by step solution

Understand the Curl Formula

Identify Components of \(\mathbf{v}\)

Compute \( \frac{\partial R}{\partial y} \) and \( \frac{\partial Q}{\partial z} \)

Compute \( \frac{\partial P}{\partial z} \) and \( \frac{\partial R}{\partial x} \)

Compute \( \frac{\partial Q}{\partial x} \) and \( \frac{\partial P}{\partial y} \)

Apply the Curl Formula

Write the Final Answer

Key Concepts

Curl of a Vector Field

Partial Derivatives

Cartesian Coordinates

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