Vector Fields Acting on a Curve

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## Vector Fields Acting on a Curve

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**VC.04-VC.08 Redux**Vector Fields Acting on a Curve**VC.04: Find a surface with the given gradient (Ex 1 From**VC.04 Notes)**VC.05: Measuring the Flow of a Vector Field ALONG a Curve**These integrals are usually called line integrals or path integrals.**VC.05: Measuring the Flow of a Vector Field Along Another**Closed Curve**VC.05: Measuring the Flow of a Vector Field Along Another**Closed Curve Counterclockwise**VC.05: Measuring the Flow of a Vector Field Across Another**Closed Curve Counterclockwise**VC.05: Path Independence: The Flow of a Gradient Field Along**an Open Curve**VC.06: Measuring the Flow of a Vector Field ALONG a Closed**Curve Let C be a closed curve with a counterclockwise parameterization. Then the net flow of the vector field ALONG the closed curve is measured by: Let region R be the interior of C. If the vector field has no singularities in R, then we can use Gauss-Green:**VC.06: Summary: The Flow of A Vector Field ALONG a Closed**Curve: • Let C be a closed curve parameterized counterclockwise. Let Field(x,y) be a vector field with no singularities on the interior region R of C. Then: • This measures the net flow of the vector field ALONG the closed curve.**VC.06: Measuring the Flow of a Vector Field ACROSS a Closed**Curve Let C be a closed curve with a counterclockwise parameterization. Then the net flow of the vector field ACROSS the closed curve is measured by: Let region R be the interior of C. If the vector field has no singularities in R, then we can use Gauss-Green:**VC.06: Summary: The Flow of A Vector Field ACROSS a Closed**Curve: • Let C be a closed curve parameterized counterclockwise. Let Field(x,y) be a vector field with no singularities on the interior region R of C. Then: • This measures the net flow of the vector field ACROSS the closed curve.**VC.06: The Divergence Locates Sources and Sinks**Let C be a closed curve with a counterclockwise parameterization with no singularities on the interior of the curve. Then:**VC.06: The Rotation Helps You Find**Clockwise/Counterclockwise Swirl Let C be a closed curve with a counterclockwise parameterization with no singularities on the interior of the curve. Then:**VC.06: Find the Net Flow of a Vector Field ACROSS Closed**Curve**VC.06: A Flow Along With Multiple Singularities? No Problem!****VC.07: Mathematica-Aided Change of Variables (Parallelogram**Region)**VC.07: Mathematica-Aided Change of Variables (Parallelogram**Region)**VC.07: Mathematica-Aided Change of Variables (Parallelogram**Region)