(1.7 hours to learn)
A surface integral is the integral of a function over a surface. Important cases include surface area and flux, where the function is the dot product of the surface normal with a vector field.
This concept has the prerequisites:
- vector fields (Flux is defined in terms of a vector field.)
- multiple integrals (Surface integrals are defined in terms of multiple integrals.)
- cross product (The definition of the surface integral includes a cross product.)
- dot product (The definition of flux includes a dot product.)
- partial derivatives (Partial derivatives are used in computing surface integrals.)
- Know the definition of a surface integral in three dimensions.
- Be able to integrate a function over a surface in three dimensions.
- As a special case, be able to compute surface area.
- Be able to compute the flux across a surface in three dimensions.
- Derive the formula for a surface integral for a surface given as z = f(x, y).
Core resources (read/watch one of the following)
→ MIT Open Courseware: Multivariable Caclulus (2010)
Video lectures for MIT's introductory multivariable calculus class.
→ Multivariable Calculus
An introductory multivariable calculus textbook.
Location: Section 15.5, "Surface integrals," pages 1047-1055
Supplemental resources (the following are optional, but you may find them useful)
→ Multivariable Mathematics
A textbook on linear algebra and multivariable calculus with proofs.
Location: Section 8.4, "Surface integrals and flux," pages 367-375
- differential forms
- This presents a more rigorous treatment than the other resources.
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