ma 242.003
DESCRIPTION
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 PresentationTRANSCRIPT
![Page 1: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/1.jpg)
MA 242.003
• Day 48 – March 21, 2013• Section 12.8: Finish: Triple Integrals in Spherical
Coordinates• Section 13.1: Vector Fields
![Page 2: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/2.jpg)
Section 12.8Triple Integrals in Cylindrical Coordinates
Spheres
Goal: Use spherical coordinates to compute a triple integral that has spherical symmetry.
Cones
![Page 3: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/3.jpg)
Fubini’s theorem for a spherical box:
![Page 4: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/4.jpg)
Generalization to non-spherical boxes
![Page 5: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/5.jpg)
Generalization to non-spherical boxes
![Page 6: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/6.jpg)
Generalization to non-spherical boxes
![Page 7: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/7.jpg)
![Page 8: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/8.jpg)
(Continuation of example)
![Page 9: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/9.jpg)
Chapter 13: Vector Calculus
![Page 10: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/10.jpg)
Chapter 13: Vector Calculus
“In this chapter we study the calculus of vector fields,
![Page 11: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/11.jpg)
Chapter 13: Vector Calculus
“In this chapter we study the calculus of vector fields,
and line integrals of vector fields (work),
![Page 12: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/12.jpg)
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,
![Page 13: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/13.jpg)
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”
![Page 14: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/14.jpg)
Section 13.1: Vector Fields
![Page 15: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/15.jpg)
Section 13.1: Vector Fields
![Page 16: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/16.jpg)
Section 13.1: Vector Fields
Wind velocity vector field 2/20/2007
![Page 17: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/17.jpg)
Section 13.1: Vector Fields
Wind velocity vector field 2/20/2007 Wind velocity vector field 2/21/2007
![Page 18: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/18.jpg)
Section 13.1: Vector Fields
Ocean currents off Nova Scotia
![Page 19: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/19.jpg)
Section 13.1: Vector Fields
Airflow over an inclined airfoil.
![Page 20: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/20.jpg)
General form of a 2-dimensional vector field
![Page 21: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/21.jpg)
General form of a 2-dimensional vector field
![Page 22: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/22.jpg)
General form of a 2-dimensional vector field
![Page 23: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/23.jpg)
General form of a 2-dimensional vector field
![Page 24: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/24.jpg)
General form of a 2-dimensional vector field
Examples:
![Page 25: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/25.jpg)
General form of a 2-dimensional vector field
Examples:
![Page 26: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/26.jpg)
General form of a 2-dimensional vector field
Examples:
![Page 27: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/27.jpg)
General form of a 2-dimensional vector field
Examples:
QUESTION: How can we visualize 2-dimensional vector fields?
![Page 28: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/28.jpg)
General form of a 2-dimensional vector field
Examples:
Question: How can we visualize 2-dimensional vector fields?
Answer: Draw a few representative vectors.
![Page 29: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/29.jpg)
Example:
![Page 30: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/30.jpg)
Example:
![Page 31: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/31.jpg)
![Page 32: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/32.jpg)
![Page 33: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/33.jpg)
![Page 34: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/34.jpg)
![Page 35: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/35.jpg)
We will turn over sketching vector fields in 3-space to MAPLE.
![Page 36: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/36.jpg)
![Page 37: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/37.jpg)
![Page 38: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/38.jpg)
![Page 39: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/39.jpg)
![Page 40: MA 242.003](https://reader035.vdocuments.us/reader035/viewer/2022062814/5681684a550346895dde397f/html5/thumbnails/40.jpg)