UL XM522 Mutivariable Integral Calculus

Transcription

1 1 UL XM522 Mutivariable Integral Calculus Instructor: Margarita Kanarsky

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3 3 Vector fields: Examples: inverse-square fields the vector field for the gravitational force

4 4 The Gradient Field:

5 5 The Divergence and the Curl of the Vector Field: a real-valued function (values: scalars) a vector-valued function (values: vectors)

6 6 The Laplacian Operator:

7 7 Line Integrals with respect to arc length: f > 0 and continuous area of the surface lying between c and the graph of f f changes sign and continuous net signed area of the surface lying between c and the graph of f

8 8 Line Integrals With Respect to Arc Length Parametrization : Line Integrals With Respect to Any Smooth Parametrization: or directly using the formula for arc length:

9 9 Other Types of Line Integrals Line Integrals With Respect to Arc Length in 3-space:. No simple geometric interpretation. Many important physical applications (the mass of a inhomogeneous wire in 3-space, work done by a force field on an object moving along a smooth curve in 3-space)

10 10 Line Integrals With Respect to x, y, and z:

11 11 Evaluating line Integrals with respect to x, y, and z: 2-space 3-space

12 12 Change of Orientation: but

13 13 Combining the different integrals under one sign:

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17 17 Line Integrals and Work Integrating a vector field along a curve:

18 18 Work as a line integral: the work performed by the force field on the particle

19 19 Integrating a vector field along a parametrized curve: for any parametrization for the arc-length parametrization Sign of work integrals: Changing orientation in work integrals:

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23 23 Conservative fields and Independence of Path: a potential function can be thought as an antiderivative of F

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26 26 Theoretical Characterization for Conservative Fields: a closed curve a connected set the initial and the terminal point coincide consists of one piece consists of multiple pieces the notions of conservative and independent of path are really equivalent characterizations of vector fields

27 27 Practical Characterization for Conservative Fields: Conservative Field Test a simple curve a simple curve does not intersect itself (except possibly at the endpoints) a simply connected set a simply connected set has no holes

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29 29 curl F = 0

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31 31 Green s Theorem for Simply Connected Regions:

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35 35 Surface Integrals: A curved lamina is modeled by a surface. The density of a lamina is given by the density function at each point. What is the mass of lamina?

36 36 How to Evaluate Surface Integrals:

37 37 Surface Integrals Over Surfaces with Equation z=g(x, y): projection on xy-plane Surface Integrals Over Surfaces with Equation y=g(x, z): projection on xz-plane Surface Integrals Over Surfaces with Equation x=g(y, z): projection on yz-plane

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39 39 Natural Orientation of a Smooth Parametric Surface: the positive orientation of the parametric surface the negative orientation of the parametric surface

40 40 Flux:

41 41 Flux: the velocity component across the k-th patch the velocity component across the k- th patch (along a normal vector) the volume of fluid across the k-th patch the net volume of fluid across the entire surface the flux of F across σ

42 42 Evaluating Flux Integrals for Parametric Surfaces:

43 43 Evaluating Flux for Non-Parametric Surfaces: Natural Parametrization Orientation of a Smooth Non-Parametric Surface:

44 44 The given surface is a level surface of G It can be shown: since:

45 45 the flux for non-parametric surfaces

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49 49 The Divergence: σ is closed, piece-wise smooth the flux of a vector field across a closed surface with the outward orientation

50 50 Divergence as a Flux Density: Consider a small spherical region G centered at P_0 with surface sigma oriented outward. the limiting flux per unit volume at a point

51 51 Sources and Sinks: Sources: Φ>0 iff divf>0 Sinks: Φ<0 iff divf<0 No sources/sinks: Φ=0 iff divf=0 (continuity equation of incompressible fluids)

52 52 Gauss s Law for Inverse-Square Fields:

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56 56 Relative Orientation of Curves and Surfaces: The surface is on person s left The surface is on person s right

57 57 Stokes Theorem: Use Stokes Theorem for calculating work around piece-wise smooth curves

58 58 Relationship between Green s Theorem and Stokes Theorem: In the 2-space:

59 59 Curl viewed as a circulation: Consider a small oriented disk σa of radius a centered at Po in a steady-state fluid flow; A(σa) is its area; Ca is its boundary; n is a unit normal vector at the center of the disk that points in the direction of orientation. Goal: Find the direction of n that produce maximum rotation rate in the positive direction of the boundary. F T contributes to rotation measures tendency to flow in the positive direction around Ca if F is continuous on σa, F(Po) F(P) for any point p on σa The circulation density of F at Po in the direction of n

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