MA 242.003 Day 51 – March 26, 2013 Section 13.1: (finish) Vector Fields Section 13.2: Line Integrals.

MA 242.003

• Day 51 – March 26, 2013• Section 13.1: (finish) Vector Fields• Section 13.2: Line Integrals

Chapter 13: Vector Calculus

Chapter 13: Vector Calculus

“In this chapter we study the calculus of vector fields,

…and line integrals of vector fields (work),

…and the theorems of Stokes and Gauss,

…and more”

Section 13.1: Vector Fields

Section 13.1: Vector Fields

Wind velocity vector field 2/20/2007

Section 13.1: Vector Fields

Wind velocity vector field 2/20/2007 Wind velocity vector field 2/21/2007

Section 13.1: Vector Fields

Ocean currents off Nova Scotia

Section 13.1: Vector Fields

Airflow over an inclined airfoil.

General form of a 2-dimensional vector field

General form of a 2-dimensional vector field

General form of a 2-dimensional vector field

Examples:

General form of a 2-dimensional vector field

Examples:

QUESTION: How can we visualize 2-dimensional vector fields?

General form of a 2-dimensional vector field

Examples:

Question: How can we visualize 2-dimensional vector fields?

Answer: Draw a few representative vectors.

Example:

Example:

Example:

We will turn over sketching vector fields in 3-space to MAPLE.

Gradient, or conservative, vector fields

Gradient, or conservative, vector fields

Gradient, or conservative, vector fields

EXAMPLES:

Gradient, or conservative, vector fields

EXAMPLES:

QUESTION: Why are conservative vector fields important?

QUESTION: Why are conservative vector fields important?

ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3)

QUESTION: Why are conservative vector fields important?

ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3)

Sections 13.2 and 13.3 are concerned with the following questions:

QUESTION: Why are conservative vector fields important?

ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3)

Sections 13.2 and 13.3 are concerned with the following questions:

1. Given an arbitrary vector field, find a TEST to apply to the vector field to determine if it is conservative.

QUESTION: Why are conservative vector fields important?

ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3)

Sections 13.2 and 13.3 are concerned with the following questions:

1. Given an arbitrary vector field, find a TEST to apply to the vector field to determine if it is conservative.

2. Once you know you have a conservative vector field, “Integrate it” to find its potential functions.

Format of chapter 13:

1. Sections 13.2, 13.3 - conservative vector fields

2. Sections 13.4 – 13.8 – general vector fields

Section 13.2: Line integrals

Section 13.2: Line integrals

GOAL: To generalize the Riemann Integral of f(x) along a line to an integral of f(x,y,z) along a curve in space.

We partition the curve into n pieces:

Assuming f(x,y) is continuous, we evaluate it at sample points, multiply by the arc length of each subarc, and form the following sum:

Assuming f(x,y) is continuous, we evaluate it at sample points, multiply by the arc length of each subarc, and form the following sum:

Assuming f(x,y) is continuous, we evaluate it at sample points, multiply by the arc length of each subarc, and form the following sum:

which is similar to a Riemann sum.

Assuming f(x,y) is continuous, we evaluate it at sample points, multiply by the arc length of each subarc, and form the following sum:

which is similar to a Riemann sum.

EXAMPLE:

Extension to 3-dimensional space

Extension to 3-dimensional space

Extension to 3-dimensional space

Shorthand notation

Extension to 3-dimensional space

Shorthand notation

Extension to 3-dimensional space

Shorthand notation

Extension to 3-dimensional space

Shorthand notation

3. Then

Line Integrals along piecewise differentiable curves