Vector Fields and Line Integrals in the Plane

John Belcher

Course Lectures

• Dot Product

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• Determinants; Cross Product

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• Matrices; Inverse Matrices

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• Square Systems; Equations of Planes

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• Parametric Equations for Lines and Curves

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• Velocity, Acceleration - Kepler's Second Law

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• Review: Vectors and Matrices

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• Level Curves; Partial Derivatives; Tangent Plane Approximation

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• Max-Min Problems; Least Squares

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• Second Derivative Test; Boundaries and Infinity

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• Differentials; Chain Rule

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• Gradient; Directional Derivative; Tangent Plane

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• Lagrange Multipliers

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• Non-Independent Variables

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• Partial Differential Equations; Review

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• Double Integrals

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• Double Integrals in Polar Coordinates; Applications

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• Change of Variables

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• Vector Fields and Line Integrals in the Plane

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• Path Independence and Conservative Fields

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• Gradient Fields and Potential Functions

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• Green's Theorem

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• Flux; Normal Form of Green's Theorem

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• Simply Connected Regions; Review

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• Triple Integrals in Rectangular and Cylindrical Coordinates

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• Spherical Coordinates; Surface Area

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• Vector Fields in 3D; Surface Integrals and Flux

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• Divergence Theorem

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• Divergence Theorem (continued): Applications and Proof

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• Line Integrals in Space, Curl, Exactness and Potentials

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• Stokes' Theorem

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• Stokes' Theorem (continued); Review

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• Topological Considerations - Maxwell's Equations

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• Multivariable Calculus Final Review

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• Multivariable Calculus Final Review (continued)

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