Line and Surface Integrals Report two key concepts in vector calculus

Line and Surface Integrals are two key concepts in vector calculus used to compute values along curves and over surfaces, respectively.

1. Line and Surface Integrals
Ayrton John V. Bantay
EgyE 612 Tensor Analysis for Engineers

2. Outline
•Line Integrals
•Surface Integrals
•Divergence Theorem
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Introduction
A line integral is an integral where the function to be
integrated is evaluated along a curve and a surface integral
is a generalization of multiple integrals to integration over
surfaces. It can be thought of as the double integral analog
of the line integral. Given a surface, one may integrate over
its scalar fields (that is, functions which return scalars as
values), and vector fields (that is, functions which return
vectors as values). Surface integrals have applications in
physics, particularly with the theories of classical
electromagnetism.

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Line Integrals

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Line Integrals

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Line Integrals

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Line Integrals

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Line Integrals

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Line Integrals

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Definition: Line Integral of a scalar Field

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Line Integrals

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Line Integrals

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Line Integrals

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Example: Line Integrals

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Example: Line Integrals

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Example: Line Integrals

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Line Integrals

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Line Integrals

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Line Integrals

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Line Integrals

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Line Integral of a Vector Field

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Line Integral of a Vector Field

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Line Integral of a Vector Field

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Line Integral of a Vector Field

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Line Integral of a Vector Field

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Surface Integral

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Surface Integral

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Surface Integral

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Surface Integral

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Surface Integral

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Surface Integral

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Surface Integral

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Surface Integral

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Definition: Surface Integral

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Example: Surface Integral

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Example: Surface Integral

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Example: Surface Integral

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Example: Surface Integral

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Surface Integral

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Surface Integral

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Definition: Surface Integral of f over Σ
surface integral of f overΣ
surface integral of f overΣ

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Surface Integral of f over Σ

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Surface Integral of f over Σ

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Surface Integral of f over Σ

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Surface Integral of f over Σ

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Surface Integral of f over Σ

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Surface Integral of f over Σ

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Divergence Theorem

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Example

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Surface Integral

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Surface Integral

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Surface Integral

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Surface Integral

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