Book cover for Calculus Early Transcendental Functions

Calculus Early Transcendental Functions

Ron Larson, Bruce Edwards

ISBN #9781285774770

6th Edition

8,973 Questions

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64,375 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Chapter 15.1 on Vector Analysis introduces vector fields and their key operations. It explains how to represent physical phenomena such as velocity, gravitational, and electric fields using vector functions. A major focus is on identifying conservative vector fields via tests on partial derivatives, finding potential functions for them, and calculating the curl and divergence. These tools not only provide insight into the geometry of the field but also link mathematical properties with physical laws like energy conservation and incompressibility.

Learning Objectives

1

Explain the concept of a vector field in both the plane and in space, including how to visualize them by sketching representative vectors.

2

Determine whether a vector field is conservative by testing conditions on its partial derivatives and, if conservative, find its potential function.

3

Calculate important vector operators such as the curl and divergence, and interpret these quantities physically.

4

Apply the methods of recovering a function from its gradient to problems in vector analysis.

5

Relate mathematical results to physical examples such as velocity fields, gravitational fields, and electric force fields.

Key Concepts

CONCEPT

DEFINITION

Vector Field

A function that assigns a vector to every point in a region of the plane or space. It is used to represent force fields, velocity fields, and other directional quantities.

Conservative Vector Field

A vector field that is the gradient of some differentiable potential function. In physical terms, the work done by forces in such fields is path-independent.

Inverse Square Field

A special type of vector field where the magnitude of the vector is inversely proportional to the square of the distance from a given point, commonly seen in gravitational and electric fields.

Curl

A vector operator that measures the rotation or swirling strength of a vector field at a given point. If the curl is zero everywhere, the field is called irrotational.

Divergence

A scalar operator that measures the magnitude of a source or sink at a given point in the vector field, indicating how much the field is spreading out or converging.

Potential Function

A scalar function whose gradient yields the original vector field. Its existence is equivalent to the field being conservative.

Example Problems

Example 1

In Exercises $1-4,$ match the vector field with its graph. [The graphs are labeled (a), (b), (c), and (d).] A.(GRAPH CAN'T COPY). B.(GRAPH CAN'T COPY). C.(GRAPH CAN'T COPY). D.(GRAPH CAN'T COPY). $$\mathbf{F}(x, y)=y \mathbf{i}$$

Example 2

In Exercises $1-4,$ match the vector field with its graph. [The graphs are labeled (a), (b), (c), and (d).] A.(GRAPH CAN'T COPY). B.(GRAPH CAN'T COPY). C.(GRAPH CAN'T COPY). D.(GRAPH CAN'T COPY). $$\mathbf{F}(x, y)=x \mathbf{j}$$

Example 3

In Exercises $1-4,$ match the vector field with its graph. [The graphs are labeled (a), (b), (c), and (d).] A.(GRAPH CAN'T COPY). B.(GRAPH CAN'T COPY). C.(GRAPH CAN'T COPY). D.(GRAPH CAN'T COPY). $$\mathbf{F}(x, y)=y \mathbf{i}-x \mathbf{j}$$

Example 4

In Exercises $1-4,$ match the vector field with its graph. [The graphs are labeled (a), (b), (c), and (d).] A.(GRAPH CAN'T COPY). B.(GRAPH CAN'T COPY). C.(GRAPH CAN'T COPY). D.(GRAPH CAN'T COPY). $$\mathbf{F}(x, y)=x \mathbf{i}+3 y \mathbf{j}$$

Example 5

In Exercises $5-10$, find $\|\mathbf{F}\|$ and sketch several representative vectors in the vector field. $$\mathbf{F}(x, y)=\mathbf{i}+\mathbf{j}$$
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Step-by-Step Explanations

QUESTION

Determine whether the vector field F(x, y) = (2x, y) is conservative.

STEP-BY-STEP ANSWER:

Step 1: Identify the component functions: let M(x, y) = 2x and N(x, y) = y.
Step 2: Compute the partial derivative of M with respect to y: M_y = ∂(2x)/∂y = 0.
Step 3: Compute the partial derivative of N with respect to x: N_x = ∂(y)/∂x = 0.
Step 4: Since M_y and N_x are equal (both 0) on the domain and the domain is simply connected, the vector field is conservative.
Final Answer: F(x, y) is conservative.

Testing for a Conservative Vector Field in the Plane

QUESTION

Find the curl of the vector field F(x, y, z) = (2xy, x^2 - z^2, 2yz).

STEP-BY-STEP ANSWER:

Step 1: Write F(x,y,z) in terms of its components: M = 2xy, N = x^2 - z^2, and P = 2yz.
Step 2: Set up the determinant for curl using the formula: curl F = | i j k; ∂/∂x ∂/∂y ∂/∂z; M N P |.
Step 3: Compute the i-component: ∂P/∂y - ∂N/∂z = (2z) - (-2z) = 4z.
Step 4: Compute the j-component: -(∂P/∂x - ∂M/∂z) = -(0 - 0) = 0.
Step 5: Compute the k-component: ∂N/∂x - ∂M/∂y = (2x) - (2x) = 0.
Step 6: Combine components to get curl F = (4z, 0, 0).
Final Answer: The curl of F is (4z, 0, 0).

Finding the Curl of a Vector Field in Space

QUESTION

Find a potential function f(x, y) for the vector field F(x, y) = (2xy, x^2).

STEP-BY-STEP ANSWER:

Step 1: Since F is conservative, there exists a function f such that f_x = 2xy and f_y = x^2.
Step 2: Integrate f_x = 2xy with respect to x: f(x, y) = ∫2xy dx = x^2y + g(y), where g(y) is an arbitrary function of y.
Step 3: Differentiate f(x, y) with respect to y: f_y = x^2 + g'(y).
Step 4: Equate f_y to x^2 (given by F): x^2 + g'(y) = x^2, which implies g'(y) = 0.
Step 5: Integrate g'(y) to get g(y) = C, where C is a constant.
Final Answer: The potential function is f(x, y) = x^2y + C.

Finding a Potential Function

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Common Mistakes

  • Assuming that every vector field is conservative without verifying the necessary conditions (such as equality of mixed partial derivatives).
  • Confusing the geometric meaning of curl with divergence; curl relates to rotation, while divergence relates to the net 'flow' outward from a point.
  • Dropping constants of integration when finding a potential function, leading to an incomplete description of the family of potential functions.
  • Incorrectly plotting vector fields by failing to choose representative sample points or using vectors of unequal magnitudes that obscure the field’s symmetry.