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MA 242.003. Day 48 – March 21, 2013 Section 12.8: Finish: Triple Integrals in Spherical Coordinates Section 13.1: Vector Fields. Section 12.8 Triple Integrals in Cylindrical Coordinates. Goal : Use spherical coordinates to compute a triple integral that has spherical symmetry. Spheres. - PowerPoint PPT Presentation
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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.