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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 ),.
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MA 242.003 • Day 51 – March 26, 2013 • Section 13.1: (finish) Vector Fields • Section 13.2: Line Integrals
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 Wind velocity vector field 2/20/2007
Section 13.1: Vector Fields Wind velocity vector field 2/21/2007 Wind velocity vector field 2/20/2007
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 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.
We will turn over sketching vector fields in 3-space to MAPLE.
Gradient, or conservative, vector fields EXAMPLES:
Gradient, or conservative, vector fields EXAMPLES:
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: 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: Given an arbitrary vector field, find a TEST to apply to the vector field to determine if it is conservative. Once you know you have a conservative vector field, “Integrate it” to find its potential functions.
Format of chapter 13: Sections 13.2, 13.3 - conservative vector fields Sections 13.4 – 13.8 – general vector fields
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.
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.
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