(Vector) Line Integrals
Dr Aiden Price
EGB241
Week 8
1/15
(Vector) Line
Integrals
EGB241
Line Integrals
Line Integrals
of Vector
Fields
Parametric
Equations
Next Week
2/15
Last Week
• Covered the processes for finding surfaces and volumes.
• It is also possible to find integrals along a line, known as
line integrals.
• To make things more complex, the focus of this lecture is
on line integrals of vector fields.
(Vector) Line
Integrals
EGB241
Line Integrals
Line Integrals
of Vector
Fields
Parametric
Equations
Next Week
3/15
Scalar Vs Vector Line Integrals
To get a better idea of the difference between the two, let’s
check out some quick animations:
(1) Check out this Wikipedia .gif on line integrals (scalar).
• Magnitude at a point on the line is the distance to the axis.
(2) Check out this Wikipedia .gif on line integrals (scalar).
• Magnitude at a point on the line is how much the vector
at that point in aligned with the direction of the line.
(Vector) Line
Integrals
EGB241
Line Integrals
Line Integrals
of Vector
Fields
Parametric
Equations
Next Week
4/15
Line Integrals of Vector Fields I
• To begin, consider a vector field in Cartesian coordinates,
with potentially changing direction at every point.
• This vector field, F, can be written as
F(x, y, z) = Fx(x, y, z)ˆ x + Fy(x, y, z)ˆ y + Fz(x, y, z)ˆ z.
• Now consider some line, C, travelling through F from
point a to point b.
(Vector) Line
Integrals
EGB241
Line Integrals
Line Integrals
of Vector
Fields
Parametric
Equations
Next Week
5/15
Line Integrals of Vector Fields II
• To find how much of F is aligned with the direction of C
at every point along C, we must integrate!
• The change along our line? The differential length!
• Remember, in Cartesian coordinates:
• dl = dx xˆ + dy yˆ + dz zˆ.
• Working with vectors – use dot products!
(Vector) Line
Integrals
EGB241
Line Integrals
Line Integrals
of Vector
Fields
Parametric
Equations
Next Week
6/15
Line Integrals of Vector Fields III
• By using dot products, some familiar ideas resurface:
ZC F · dl = ZC Fxxˆ · dl + ZC Fyyˆ · dl + ZC Fzzˆ · dl
= ZC Fxxˆ · (dx xˆ) + ZC Fyyˆ · (dy yˆ) + ZC Fzzˆ · (dz zˆ)
= ZC Fxdx + ZC Fydy + ZC Fzdz
• This equation will only work, however, if C changes in
only one direction at a time.
• What about curved, diagonal or otherwise “squiggly” lines?
(Vector) Line
Integrals
EGB241
Line Integrals
Line Integrals
of Vector
Fields
Parametric
Equations
Next Week
7/15
Parameterisation
• When the line traverses more than one dimension at once,
parameterisation is used to express F as a series of
implicit equations, with independent variable t (recall
MXB125).
• Mathematically, we replace parts of F as follows:
F(x(t), y(t), z(t)) = F(r(t)).
• Which, when integrated (derivation excluded) gives
ZC F · dr = Zab F(r(t)) · r0(t) dt.
(Vector) Line
Integrals
EGB241
Line Integrals
Line Integrals
of Vector
Fields
Parametric
Equations
Next Week
8/15
Parameterisation II
• There are infinitely many lines with infinitely many shapes.
• In EGB241, consider straight lines and perfectly round
lines.
• Mainly for ease; there are plenty more plausible curves but
none so easy to consider as a straight or circular line.
(Vector) Line
Integrals
EGB241
Line Integrals
Line Integrals
of Vector
Fields
Parametric
Equations
Next Week
9/15
Parameterisation – Straight
• Think of a straight line as one having an intercept and a
slope.
• If our line is represented solely as a function of t, then it
essentially becomes a linear line (albeit with x, y and z
components).
• Treat one point as the starting point and another as the
end point. Then the starting point, c1, acts as our
intercept (when t = 0) and the difference from the start to
the end point, c2 – c1, is our slope.
r(t) = (c2 – c1)t + c1, 0 ≤ t ≤ 1.
(Vector) Line
Integrals
EGB241
Line Integrals
Line Integrals
of Vector
Fields
Parametric
Equations
Next Week
10/15
Parameterisation – Example
• Calculate the integral of a line on a vector field, where
• F(x, y, z) = xz xˆ – yz zˆ, and
• C is the line segment from (-1, 2, 0) to (3, 0, 1).
(Vector) Line
Integrals
EGB241
Line Integrals
Line Integrals
of Vector
Fields
Parametric
Equations
Next Week
11/15
Parameterisation – Circular
• Consider a circular line in Cartesian coordinates.
• There is no change in height, assuming the circle is flat, so
the z component can be ignored.
• What if we consider cylindrical coordinates?
• There is no height, so z is not a factor. There is no
change in radius, as our circle is centered correctly.
• Therefore, F in cylindrical coordinates will have only one
difference length, dlφ.l
(Vector) Line
Integrals
EGB241
Line Integrals
Line Integrals
of Vector
Fields
Parametric
Equations
Next Week
12/15
Line Integral – Example I
• Ampere’s law: The line integral of the magnetic field, H,
along a closed path, C, is equal to the current, I, passing
through the surface enclosed by the path. Written
mathematically:
IC H · dl = I,
where H represents a closed line integral.
• Consider an infinite long conductor aligned along the
z-axis, carrying a steady current, I.
(1) Draw a simple diagram illustrating this line integral.
(2) Determine the magnetic field, H, due to the current, I.
(Vector) Line
Integrals
EGB241
Line Integrals
Line Integrals
of Vector
Fields
Parametric
Equations
Next Week
13/15
Line Integral – Example II
• Consider now an electric field, E, which has a potential
difference between two points of Vp1,p2 = V1 – V2, or
Vp
1,p2 = – Zp2p1 E · dl.
• Consider a parallel plate capacitor with an electric field
E = -1000V/m2 zˆ and a plate separation of d = 1mm.
(1) Draw a simple diagram illustrating a line integral from a
point p1 in the first plate to a point p2 in the second plate.
(2) Calculate the potential difference between the plates.
(Vector) Line
Integrals
EGB241
Line Integrals
Line Integrals
of Vector
Fields
Parametric
Equations
Next Week
14/15
Line Integral – Example III
• Use this image to help answer the following questions:
• Given a vector function F = xyxˆ + (3x – y2)2yˆ, evaluate
the line integral R F · dl from point P1 = (5, 6) to
P2 = (3, 3) along
(1) The direct path, P1P2,
(2) and along the path P1AP2, where A = (5, 3).
(Vector) Line
Integrals
EGB241
Line Integrals
Line Integrals
of Vector
Fields
Parametric
Equations
Next Week
15/15
Next Week
• Gradients.
• Divergence.
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