Chapter 4: Problem 20
Find the curl, \(\nabla \times \mathbf{F},\) for the following forces: \((\mathbf{a}) \mathbf{F}=k \mathbf{r} ;(\mathbf{b}) \mathbf{F}=\left(A x, B y^{2}, C z^{3}\right) ;(\mathbf{c}) \mathbf{F}=\) \(\left(A y^{2}, B x, C z\right),\) where \(A, B, C\) and \(k\) are constants.
Short Answer
Expert verified
A: \(0 ;\) B: \(0 ;\) C: \((B-C, 0, -2Ay).\)
Step by step solution
01
Understanding the Curl
The curl of a vector field \( \mathbf{F} \) is calculated using the operator \( abla \times \mathbf{F} \). For a vector field \( \mathbf{F} = (F_x, F_y, F_z) \), the curl is given by the determinant: \[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ F_x & F_y & F_z \end{vmatrix} \] where \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are the unit vectors in the \( x, y, \) and \( z \) directions.
02
Evaluating Curl for \( \mathbf{F} = k \mathbf{r} \) (Part A)
For part (a), \( \mathbf{F} = k \mathbf{r} = (kx, ky, kz) \). Calculate the determinant: \[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ kx & ky & kz \end{vmatrix} \]. Solve this, and you'll find that each element of the resultant vector becomes zero: \( 0 \).
03
Evaluating Curl for \( \mathbf{F} = (Ax, By^2, Cz^3) \) (Part B)
For part (b), \( \mathbf{F} = (Ax, By^2, Cz^3) \). Calculate: \[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ Ax & By^2 & Cz^3 \end{vmatrix} \]. After computing the partial derivatives, the result is \( (0, 0, 0) \), showing the components cancel or produce zero.
04
Evaluating Curl for \( \mathbf{F} = (Ay^2, Bx, Cz) \) (Part C)
For part (c), \( \mathbf{F} = (Ay^2, Bx, Cz) \). Compute: \[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ Ay^2 & Bx & Cz \end{vmatrix} \]. Upon solving, the resulting vector is \( (B-C, 0, -2Ay) \), indicating non-zero curl components.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Vector Fields
Vector fields are vital in physics and mathematics, especially when studying fluid dynamics and electromagnetic fields. A vector field assigns a vector to every point in space. This vector can represent anything from velocity in a fluid to the force at different points within an electric field.
For example:
For example:
- In a fluid, each small volume element might have a vector describing its velocity, which collectively form a velocity field.
- In electromagnetism, an electric field at a point can be described by a vector pointing in the direction of the force experienced by a positive test charge at that point.
Introduction to Partial Derivatives
Partial derivatives play a significant role in multivariable calculus, especially in fields like physics and engineering where functions depend on multiple variables. When dealing with a function of two or more variables, the partial derivative measures how the function changes as one variable changes, keeping others constant.
For instance, consider a temperature field in a room, which might depend on three spatial variables: temperature changes can be assessed by differentiating with respect to one spatial dimension while keeping the others fixed.
In formulas:
For instance, consider a temperature field in a room, which might depend on three spatial variables: temperature changes can be assessed by differentiating with respect to one spatial dimension while keeping the others fixed.
In formulas:
- The partial derivative of a function \( f(x, y) \) with respect to \( x \) is denoted \( \frac{\partial f}{\partial x} \).
- It shows the rate of change in the \( x \)-direction.
- To find the partial derivatives for \( \mathbf{F} = (Ax, By^2, Cz^3) \), we look at each component. For example, \( \frac{\partial (By^2)}{\partial x} = 0 \) since \( By^2 \) does not change with \( x \).
Determinant of a Matrix and its Use in Curl Calculation
The determinant of a matrix is a unique number associated with a square matrix. It's a fundamental concept used in various applications, including solving systems of linear equations and transforming vector spaces. In the context of vector calculus, determinants play a crucial role in operations like finding the curl of a vector field.
To calculate the curl of a vector field \( \mathbf{F} = (F_x, F_y, F_z) \), we use a special type of determinant called the Jacobian determinant. It involves a vector component matrix and a variable's partial derivatives:
To calculate the curl of a vector field \( \mathbf{F} = (F_x, F_y, F_z) \), we use a special type of determinant called the Jacobian determinant. It involves a vector component matrix and a variable's partial derivatives:
- The determinant for curl: \( abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ F_x & F_y & F_z \end{vmatrix} \)
- This 3x3 matrix involves unit vectors and partial derivatives, combined to give the curl's components.