Measure how a vector field spreads out and rotates; appear in fluid and electromagnetic equations.
Divergence ∇·F measures net outflow per unit volume (sources and sinks); curl ∇×F measures rotation. An incompressible fluid has zero divergence; an irrotational flow has zero curl. Maxwell's equations are written with them.
F=(x,y,0) has divergence 2 and zero curl.
These operators describe how fields expand, rotate and source flows, and they appear directly in the equations of fluids and electromagnetism.
Measure how a vector field spreads out and rotates; appear in fluid and electromagnetic equations.
These operators describe how fields expand, rotate and source flows, and they appear directly in the equations of fluids and electromagnetism.
It measures net outflow from a point. A positive value means the field is spreading out, like a source, and a negative value means it is converging, like a sink.
When its curl is zero everywhere, the field is irrotational and, in simply connected regions, can be written as the gradient of a potential.
Typical applications include checking whether a velocity field is incompressible using divergence, computing the vorticity of a rotating flow with curl and interpreting Maxwell's equations for a charged sphere.
Study "Line and surface integrals" first (topic 45 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Divergence is a scalar. Curl is a vector. ∇×(∇f)=0 and ∇·(∇×F)=0.
F=(x,y,0) has divergence 2 and zero curl.
The next topic is "Green, Stokes and divergence theorems", which builds directly on this one.
It is classed as Intermediate in Engineering Mathematics (topic 46 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.
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