Book cover for Calculus: Early Transcendentals

Calculus: Early Transcendentals

James Stewart

ISBN #9781285741550

8th Edition

6,422 Questions

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2,819,387 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces vector fields in both R² and R³, exploring how these fields are represented, visualized, and used in real-world situations such as wind patterns, fluid flow, gravitational and electric forces. It also covers gradient fields, highlighting that the gradient of a scalar function forms a vector field perpendicular to its level curves, and touches on the concept of conservative fields. Additionally, the section explains the setup and evaluation of line integrals over curves, emphasizing their applications in various fields of science and engineering.

Learning Objectives

1

Understand the definitions and representations of vector fields in R² and R³, including how they are expressed in terms of component functions.

2

Differentiate between velocity fields and force fields and discuss real-world examples such as wind patterns, ocean currents, gravitational and electric fields.

3

Develop skills in sketching vector fields by computing representative vectors and recognizing patterns like tangency to geometric curves.

4

Comprehend the concept of gradient vector fields, including how gradients relate to level curves and the idea of conservative fields.

5

Learn how to set up and evaluate line integrals over parameterized curves.

Key Concepts

CONCEPT

DEFINITION

Vector Field

A function that associates a vector to every point in a region of R² or R³. It is often represented in component form, e.g., F(x, y) = P(x, y)i + Q(x, y)j in R².

Velocity Field

A type of vector field that represents the velocity (magnitude and direction) of a moving fluid or air at every point. Examples include wind patterns or fluid flow inside a pipe.

Force Field

A vector field that assigns a force vector to every point in space. Typical examples are gravitational and electric fields.

Gradient Vector Field

The vector field obtained by taking the gradient (∇f) of a scalar function f. The vectors are perpendicular to the level curves of f and their length represents the rate of change of f.

Conservative Field

A vector field that is the gradient of some scalar potential function. In a conservative field, the line integral between any two points is independent of the path taken.

Line Integral

An integral where a function is integrated along a curve, summing contributions of f(x, y) multiplied by differential arc length ds. It extends the concept of integration from intervals to curves.

Example Problems

Example 1

Sketch the vector field $ \textbf{F} $ by drawing a diagram like Figure 5 or Figure 9. $ \textbf{F} (x, y) = 0.3 \textbf{i} - 0.4 \textbf{j} $

Example 2

Sketch the vector field $ \textbf{F} $ by drawing a diagram like Figure 5 or Figure 9. $ \textbf{F} (x, y) = \frac{1}{2}x \textbf{i} + y \textbf{j} $

Example 3

Sketch the vector field $ \textbf{F} $ by drawing a diagram like Figure 5 or Figure 9. $ \textbf{F} (x, y) = -\frac{1}{2} \textbf{i} + (y - x) \textbf{j} $

Example 4

Sketch the vector field $ \textbf{F} $ by drawing a diagram like Figure 5 or Figure 9. $ \textbf{F} (x, y) = y \textbf{i} + (x + y) \textbf{j} $

Example 5

Sketch the vector field $ \textbf{F} $ by drawing a diagram like Figure 5 or Figure 9. $ \textbf{F} (x, y) = \dfrac{y \textbf{i} + x \textbf{j}}{\sqrt{x^2 + y^2}} $
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Step-by-Step Explanations

QUESTION

Sketch the vector field F(x, y) = -2y i + x j by computing representative vectors and identifying any geometric patterns.

STEP-BY-STEP ANSWER:

Step 1: Choose a set of representative points. For example, at (1, 0): F(1, 0) = -2(0)i + 1j = (0, 1).
Step 2: Evaluate at another point, say (0, 1): F(0, 1) = -2(1)i + 0j = (-2, 0).
Step 3: Compute F at additional points to see the trend. At (-1, 0), F(-1, 0) = -2(0)i + (-1)j = (0, -1); at (0, -1), F(0, -1) = -2(-1)i + 0j = (2, 0).
Step 4: Recognize that the computed vectors are always perpendicular to the position vectors (x, y), showing that each vector is tangent to a circle centered at the origin.
Final Answer: The vector field F(x, y) = -2y i + x j produces vectors that are tangent to circles centered at the origin, indicating a counterclockwise rotation.

Sketching a Vector Field

QUESTION

Evaluate the line integral of f(x, y) = x² + y² along the curve C parameterized by r(t) = (cos t, sin t) for t in [0, π].

STEP-BY-STEP ANSWER:

Step 1: Express f in terms of t using the parameterization. Since x = cos t and y = sin t, f(r(t)) = cos²t + sin²t, which simplifies to 1.
Step 2: Compute the derivative r'(t) = (-sin t, cos t) and find its magnitude |r'(t)| = √(sin²t + cos²t) = 1.
Step 3: Set up the line integral using the formula: ∫C f(x, y) ds = ∫(from t = 0 to π) f(r(t)) |r'(t)| dt.
Step 4: Substitute the computed values: ∫₀^π 1 * 1 dt = ∫₀^π dt = π.
Final Answer: The value of the line integral ∫C f(x, y) ds is π.

Evaluating a Line Integral

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Common Mistakes

  • Confusing the magnitude of vectors with their directional properties, leading to errors in sketching vector fields.
  • Failing to sample enough representative points, which can obscure the overall pattern of the vector field.
  • Overlooking the need for the continuity of component functions when analyzing vector fields.
  • Assuming that all vector fields are conservative without verifying the existence of a potential function.
  • Neglecting proper parameterization or miscalculating the derivative magnitude when evaluating line integrals.