IntermediateTopic 47 of 104Engineering Mathematics
Green, Stokes and divergence theorems
Convert between boundary and interior integrals; the base of Maxwell equations and conservation laws.
In depth
Green's theorem relates a line integral around a closed curve to a double integral over the region; Stokes relates circulation to the curl through a surface; the divergence theorem equates flux out of a closed surface to the integral of divergence inside. They convert between boundary and interior forms.
Key points
- Gauss's law from the divergence theorem
- Faraday's law from Stokes
- Orientation conventions matter
ExampleTotal charge inside a closed surface equals ε₀ times the flux of E.
What is Green, Stokes and divergence theorems?
Convert between boundary and interior integrals; the base of Maxwell equations and conservation laws.
Why is it important to understand green, Stokes and divergence theorems?
These theorems connect integrals over a region to integrals over its boundary, which is how conservation laws are turned into usable equations.
What does the divergence theorem say?
The total outflow through a closed surface equals the integral of the divergence over the enclosed volume. It is a conservation statement in mathematical form.
Why does Stokes theorem matter?
It relates the circulation around a closed loop to the flux of curl through any surface bounded by that loop, which is used in electromagnetism and fluid vorticity.
How is Green, Stokes and divergence theorems used in practice?
Typical applications include converting a volume balance of heat into a boundary flux calculation, verifying a flow rate by integrating over a closed surface and deriving Maxwell's equations in integral form for a practical sensor.
What should I learn before Green, Stokes and divergence theorems?
Study "Divergence and curl" first (topic 46 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Green, Stokes and divergence theorems?
Gauss's law from the divergence theorem. Faraday's law from Stokes. Orientation conventions matter.
Can you give an example of Green, Stokes and divergence theorems?
Total charge inside a closed surface equals ε₀ times the flux of E.
What comes after Green, Stokes and divergence theorems?
The next topic is "Matrices", which builds directly on this one.
How difficult is Green, Stokes and divergence theorems?
It is classed as Intermediate in Engineering Mathematics (topic 47 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.