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MA 242.003 Day 51 – March 26, 2013 Section 13.1: (finish) Vector Fields Section 13.2: Line Integrals
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Chapter 13: Vector Calculus
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“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”
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Section 13.1: Vector Fields
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Wind velocity vector field 2/20/2007
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Section 13.1: Vector Fields Wind velocity vector field 2/20/2007 Wind velocity vector field 2/21/2007
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Section 13.1: Vector Fields Ocean currents off Nova Scotia
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Section 13.1: Vector Fields Airflow over an inclined airfoil.
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General form of a 2-dimensional vector field
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Examples:
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General form of a 2-dimensional vector field Examples: QUESTION: How can we visualize 2-dimensional vector fields?
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General form of a 2-dimensional vector field Examples: Question: How can we visualize 2- dimensional vector fields? Answer: Draw a few representative vectors.
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Example:
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We will turn over sketching vector fields in 3- space to MAPLE.
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Gradient, or conservative, vector fields
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EXAMPLES:
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Gradient, or conservative, vector fields EXAMPLES:
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QUESTION: Why are conservative vector fields important?
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ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3)
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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:
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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.
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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.
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Format of chapter 13: 1.Sections 13.2, 13.3 - conservative vector fields 2.Sections 13.4 – 13.8 – general vector fields
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Section 13.2: Line integrals
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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.
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We partition the curve into n pieces:
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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:
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which is similar to a Riemann sum.
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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.
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EXAMPLE:
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Extension to 3-dimensional space
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Shorthand notation
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Extension to 3-dimensional space Shorthand notation
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Extension to 3-dimensional space Shorthand notation
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Extension to 3-dimensional space Shorthand notation 3. Then
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Line Integrals along piecewise differentiable curves
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