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Chapter 18: Problem 25

Verify the identity. $$ \nabla \times(f \mathbf{F})=f(\nabla \times \mathbf{F})+(\nabla f) \times \mathbf{F} $$

Short Answer

Expert verified
The identity is verified by expanding both sides and showing equality through term-by-term comparison.

Step by step solution

01

Understand the Exercise Context

We need to verify a vector calculus identity involving the cross product of a scalar function \( f \) and a vector field \( \mathbf{F} \). This involves applying vector calculus operations like the gradient and curl.
02

Recall Basic Definitions

The definitions are crucial: for a vector field \( \mathbf{F} = (F_1, F_2, F_3) \) and a scalar field \( f \), the gradient is \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \) and the curl is \( abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \).
03

Express \( \nabla \times (f \mathbf{F}) \) in Terms of Components

Apply the curl operation to the vector field \( f \mathbf{F} = (f F_1, f F_2, f F_3) \): \[abla \times (f \mathbf{F}) = \left( \frac{\partial (fF_3)}{\partial y} - \frac{\partial (fF_2)}{\partial z}, \frac{\partial (fF_1)}{\partial z} - \frac{\partial (fF_3)}{\partial x}, \frac{\partial (fF_2)}{\partial x} - \frac{\partial (fF_1)}{\partial y} \right)\]
04

Use Product Rule for Partial Derivatives

Apply the product rule to differentiate terms like \( \frac{\partial (fF_3)}{\partial y} = \frac{\partial f}{\partial y} F_3 + f \frac{\partial F_3}{\partial y} \). Use similar rules for all terms obtained in Step 3.
05

Compare With the Right-Hand Side

The right side of the identity is \[ f(abla \times \mathbf{F}) + (abla f) \times \mathbf{F} \]Calculate each part:1. \( f(abla \times \mathbf{F}) = f \left( \begin{array}{c} \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \ \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \ \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \end{array} \right) \).2. \( (abla f) \times \mathbf{F} = abla f \left( \mathbf{i}, \mathbf{j}, \mathbf{k} \right) \times \mathbf{F} (F_1, F_2, F_3) \).
06

Combine Terms and Verify Equality

By following through the differentiations in Step 4 and comparing each term with those in the right-hand side expression, verify that both expressions (Step 3's result and Step 5's right-hand sum) are the same.

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Key Concepts

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

Cross Product
The cross product is a binary operation on two vectors in three-dimensional space. It produces a vector that is perpendicular to both of the original vectors. This operation is commonly used in vector calculus and physics to find a vector orthogonal to a plane.
  • To compute the cross product of two vectors \( \mathbf{A} = (A_1, A_2, A_3) \) and \( \mathbf{B} = (B_1, B_2, B_3) \), the formula to use is:
    \[ \mathbf{A} \times \mathbf{B} = (A_2B_3 - A_3B_2, A_3B_1 - A_1B_3, A_1B_2 - A_2B_1) \]
  • The resulting vector's direction is determined using the right-hand rule, a visual mnemonic where the thumb indicates the direction of the resultant vector when your fingers curl in the direction from the first to the second vector.
  • The magnitude of the cross product gives the area of the parallelogram formed by the two vectors.
The cross product is utilized in various applications such as computing torques, understanding magnetic forces, and in this exercise, verifying identities involving vector fields and scalar functions.
Gradient
The gradient is a vector operation that assigns a vector to each point in the domain of a scalar field. The gradient points in the direction of the greatest rate of increase of the function and its magnitude is the rate of change.
  • For a scalar field \( f(x, y, z) \), the gradient is denoted as \( abla f \) and calculated as:
    \[ abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \]
  • This vector points towards the steepest ascent of the function at any given point.
  • The gradient is especially useful in physics and engineering for understanding how physical quantities change in space, and plays a crucial role in the discussed identity involving vector calculus.
Curl
In vector calculus, the curl is an operator that describes the infinitesimal rotation of a vector field. Imagine tiny whirlpools or spirals of fluid rotation that measure the tendency of the field to rotate around a point.
  • For a vector field \( \mathbf{F}(x, y, z) = (F_1, F_2, F_3) \), the curl is given by:
    \[ abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \]
  • It results in a vector that represents the axis of rotation and the degree of the twisting in the field.
  • Curl is particularly relevant to electromagnetic theory and fluid dynamics, where it explains how vector fields circulate and interact.
In this exercise, we apply the curl to verify the identity, showcasing its role in describing complex interactions between scalar functions and vector fields.
Scalar Field
A scalar field assigns a single scalar value to every point in space. Think of it as a landscape where each point has a certain height. This height represents a physical quantity that varies from point to point.
  • In mathematical terms, a scalar field \( f \) provides a function \( f(x, y, z) \) mapping points in space to scalar values like temperature, pressure, or density.
  • Scalar fields are often visualized using contour plots or level surfaces displaying the function's value across different layers.
  • The primary interest in scalar fields in vector calculus is understanding how they interact with vector fields through operations like gradients and products.
Scalar fields are essential components in the identity we are verifying, influencing how the gradient couples with the vector field and the overall dynamical behavior of the system.
Vector Field
A vector field is a function that assigns a vector to every point in space. Imagine each point in space having an arrow pointing in a certain direction with a specific length. This representation is vital for describing various physical phenomena.
  • Mathematically, a vector field \( \mathbf{F} \) in three dimensions is represented as \( \mathbf{F}(x, y, z) = (F_1, F_2, F_3) \), where each component \( F_i \) is a function of \( x, y, \) and \( z \).
  • Vector fields are used to describe flow fields in fluids, electromagnetic fields, and any system where force and direction are defined at every point.
  • Understanding how vector fields interact with scalar fields and under operations like curl is crucial in solving and verifying identities in vector calculus.
In our exercise, vector fields play a central role as they interact with scalar fields and are manipulated through operations like the curl to understand dynamic spatial behaviors.

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Most popular questions from this chapter

Use Green's theorem to evaluate the line integral $$\oint_{c} x y d x+\left(x^{2}+y^{2}\right) d y$$ if \(C\) is the given curve. \(C\) is the circle \(x^{2}+y^{2}-2 x=0\).

A surface (or triple) integral of a vector function is defined as the sum of the surface (or triple) integrals of each component of the function. Using this fact and assuming that \(S\) and \(Q\) satisfy the conditions of the divergence theorem, establish the result. \(\iint_{S} f \mathbf{n} d S=\iiint_{Q} \nabla f d V\)

Use the divergence theorem (18.26) to find the flux of F through \(S\). \(\mathbf{F}(x, y, z)=3 x \mathbf{i}+x z \mathbf{j}+z^{2} \mathbf{k}\) \(S\) is the surface of the region bounded by the paraboloid \(z=4-x^{2}-y^{2}\) and the \(x y\) -plane.

If \(S\) is given by \(x=k(y, z)\) and \(\mathbf{F}=M \mathbf{i}+N \mathbf{j}+P \mathbf{k}\), show that $$ \iint_{S} \mathbf{F} \cdot \mathbf{n} d S=\iint_{R_{y z}}\left[M-N k_{y}(y, z)-P k_{z}(y, z)\right] d y d z $$

Assume that \(S\) and \(Q\) satisfy the conditions of the divergence theorem. If \(\mathbf{F}\) has continuous second partial derivatives, prove that \(\iint_{S} \operatorname{curl} \mathbf{F} \cdot \mathbf{n} d S=0\).
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