Electricity and Magnetism

Mathematical Tools and Fields

01Series Approximations: Keeping One or Two Terms 02Integrals as Sums: Along a Line and a Curved Path 03Line Integrals: Work as the Integral of F · ds 04Closed Line Integrals: Work Around a Loop 05Double Integrals: Area, Charge and Volume 06Closed Surface Integrals: The Sphere and Its Charge 07Electric Flux Through Surfaces and Volume Integrals Problem set0/10 Problem set 20/10 MIT problem set0/4

01Field Theory Versus Action at a Distance 02Scalar Fields: Contour, Color and Relief Maps 03Vector Fields, Sources, Sinks and Divergence 04Curl: Measuring the Circulation of a Field 05Gravitational, Electric and Magnetic Fields 06Drawing Vector Fields: Arrows, Lines and Grass Seeds Problem set0/10 Problem set 20/10 MIT problem set0/1

Vector Fields, Sources, Sinks and Divergence

How does the divergence of a velocity field tell a source from a sink at a single point?

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Your summary note

1. A vector field and its scalar components
  Define a vector field as a magnitude and direction at every point, write $\vec{F}=F_x\hat{i}+F_y\hat{j}+F_z\hat{k}$ with each component a scalar field, and use a fluid's velocity field as the example.
2. Sources and sinks in fluid flow
  Sketch a diverging flow from a source and a converging flow into a sink, add one combination such as a source in a uniform downward flow, and state what the texture streaks encode.
3. The divergence test $\nabla\cdot\vec{v}$
  Write the del operator and $\nabla\cdot\vec{v}=\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z}$ , then record the sign conditions: positive at a source, negative at a sink, zero at neither.
Attempt 1 of 2