MA 242.003

• Day 48 – March 21, 2013• Section 12.8: Finish: Triple Integrals in Spherical

Coordinates• Section 13.1: Vector Fields

Section 12.8Triple Integrals in Cylindrical Coordinates

Spheres

Goal: Use spherical coordinates to compute a triple integral that has spherical symmetry.

Cones

Fubini’s theorem for a spherical box:

Generalization to non-spherical boxes

Generalization to non-spherical boxes

Generalization to non-spherical boxes

(Continuation of example)

Chapter 13: Vector Calculus

Chapter 13: Vector Calculus

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

Chapter 13: Vector Calculus

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

and line integrals of vector fields (work),

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,

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

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

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:

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:

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