Line Integrals Properties/Notation The line integral is sometimes - - PowerPoint PPT Presentation

Line Integrals – Properties/Notation

◮ The line integral is sometimes denoted as

• C

F · ds =

• C

F1dx + F2dy + F3dz for F = (F1, F2, F3).

◮ If C is simple curve and consists of a union of simple smooth curves

C1, C2, ... Cn, we have

• C

F · ds =

• C1

F · ds +

• C2

F · ds + · · · +

• Cn

F · ds

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Line Integral of a Gradient

If f is differentiable a scalar function and C is a smooth curve then

• C

∇f · ds = f (r(b)) − f (r(a)) A vector field F such that F = ∇f for some differentiable scalar function f , is called conservative vector field and the function f is called potential

• f F.

Example: The gravitational field of a mass M, reads g = −GM r3 r where r = (x, y, z) is conservative, as the function V = GM r is a potential of g.

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Line Integral of Conservative Fields

Let F = ∇f conservative field. The identity

• C

∇f · ds = f (r(b)) − f (r(a)) highlights the following: BIG IDEA: The value of the line integral of a conservative vector field is independent of the path C! It only depends on the value of the potential at the initial and final points. If r(a) = r(b) then

• C

∇f · ds = 0 for any simple smooth curve C.

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Parametrisation of surfaces

A simple curve in space can be parameterised by a vector function of one variable r(t) = (x(t), y(t), z(t)) a ≤ t ≤ b. How do we parameterise a surface? Consider the sphere of radius 1 centered at the origin. In spherical coordinates it can be described as r = 1, 0 ≤ θ < 2π, 0 ≤ φ ≤ π

• r (formally) as

r(θ, φ) = (cos θ sin φ, sin θ sin φ, cos φ) BIG IDEA: A (simple & smooth) surface can be parameterised by a vector function of 2 variables!

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Parametrisation of surfaces

Thus, in general, a surface S can be parameterised as r(u, v) = (x(u, v), y(u, v), z(u, v)), a ≤ u ≤ b, c ≤ v ≤ d. Example: What is a parametrisation of a cylinder of radius 2 and height 2, whose lower basis lies in the xy-plane? Answer: Cylindrical Coordinates: r(u, v) = (2 cos u, 2 sin u, v) 0 ≤ u < 2π and 0 ≤ v ≤ 2

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