Line Integrals of Vector Fields

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Vector Line Integrals

Based on “Vector Line Integrals” in section 5.2 of the textbook.

Key Ideas

A line integral of a vector field is a “total” of the components of the field parallel to the path.

Calculation formula(s):

Properties of vector line integrals (see the “Properties of Vector Line Integrals” theorem in the textbook).

Reversing the orientation of the path changes the sign of a vector line integral.

Examples

Consider a simple radial field, F(x, y) = ⟨ x, y ⟩ , and two paths r(t) = ⟨ cos t, sin t ⟩ for 0 ≤ t ≤ 2π and s(t) = ⟨ t, 1 ⟩ for 0 ≤ t ≤ 1. What are the integrals of F along each of r and s?

Start with the integral along r, and use the “F( r(t) ) • r′(t)” calculation formula. This formula means to use the components of r(t) as x and y in F, then take the dot product with the derivative of r. Then simplify and integrate…

Notice that the integral is 0, consistent with the realization that the vector field is always perpendicular to the circular path, and so has no component parallel to it.

Calculate the integral along the second part similarly. This time the integral is non-0, reflecting the fact that the vectors from the field generally have non-0 projections onto line s:

Try a 3-dimensional example, with the integral in the “P dx …” form:

Once you phrase the components of the vector field, and the components of the derivative of r, as functions of t, everything in the integral is in terms of t and you can integrate as a single-variable integral:

Next

Flux line integrals (and circulation).

Please read “Flux” and “Circulation” in section 5.2 of the textbook.

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