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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.04: Gradient Fields. VC.04: Gradient Fields. VC.04: Breaking Field Vectors into Two Components. VC.04: Flow Along and Flow Across.
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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)