Download An Introduction to a Line Integral of a Vector Field

Introduction to a Line Integral of a Vector Field
Math Insight
A line integral (sometimes called a path integral) is the integral of some function
along a curve. One can integrate a scalar-valued function1 along a curve,
obtaining for example, the mass of a wire from its density. One can also integrate
a certain type of vector-valued functions along a curve. These vector-valued
functions are the ones where the input and output dimensions are the same, and
we usually represent them as vector fields2.
One interpretation of the line integral of a vector field is the amount of work that a
force field does on a particle as it moves along a curve. To illustrate this concept,
we return to the slinky example3 we used to introduce arc length. Here, our slinky
will be the helix parameterized4 by the function c(t)=(cost,sint,t/(3π)), for 0≤t≤6π,
Imagine that you put a small-magnetized bead on your slinky (the bead has a
small hole in it, so it can slide along the slinky). Next, imagine that you put a large
magnet to the left of the slink, as shown by the large green square in the below
applet. The magnet will induce a magnetic field F(x,y,z), shown by the green
arrows.
Particle on helix with magnet. The red helix is parametrized by c(t)=(cost,sint,t/(3π)), for 0≤t≤6π.
For a given value of t (changed by the blue point on the slider), the magenta point represents a
magnetic bead at point c(t). The green rectangle represents a large magnet, which induces the
constant magnetic field represented by the vector field F(x,y,z)=(−1/2,0,0)and illustrated with the
green arrows. The magnetic field does work on the bead as it moves along the curve.
Source URL: http://mathinsight.org/line_integral_vector_field_introduction
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
www.saylor.org
Page 1 of 5
Since the bead is magnetized, the magnetic field exerts a force on the bead.
However, we'll assume the bead's position is constrained be at c(t). We'll ignore
any effects of the magnetic force on the position of the bead.
As you change t to move the bead, work is done by the magnetic field on the
bead. Work equals forces times distance. If the magnetic field were aligned with
the direction of movement, calculating the work would be simple.
However, the bead does not move in the direction of the magnetic field. It is
constrained to move along the slinky. The work is calculated from the component
of the force in the direction of movement. For example, when the movement is 90
degrees from the direction of the force, the magnetic field does no work.
First, what is the direction of movement? It is the direction given by the
velocity5 c′(t). Denote by T the unit vector6 in the direction of
movement: T(t)=c′(t)/∥c′(t)∥. Since c′(t) (and therefore T) is tangent to the path,
we refer to T as the unit tangent vector. It is shown by the blue vector in the
below figure.
Particle on helix with magnet and tangent vector. The red helix is parameterized
by c(t)=(cost,sint,t/(3π)), for 0≤t≤6π. For a given value of t (changed by the blue point on the
slider), the magenta point represents a magnetic bead at point c(t). The blue vector represents
the tangent vector in the direction of the particle's movement for increasing t. The green rectangle
represents a large magnet, which induces the constant magnetic field represented by the vector
field F(x,y,z)=(−1/2,0,0)and illustrated with the green vector. The work done by the magnetic field
on the particle is determined by the component of F in the direction of the tangent vector, which is
shown by the cyan mark on the green slider.
The component of the force in the direction of movement is simply the dot
product of the force F at the point c(t) with the tangent vector T(t), i.e., it
Source URL: http://mathinsight.org/line_integral_vector_field_introduction
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
www.saylor.org
Page 2 of 5
is F(c(t))⋅T(t). For shorthand, we'll write it as F⋅T. This number is displayed by the
cyan mark on the green slider. Recall that the dot product is zero if the two
vectors are orthogonal, is negative if their angle is greater than 90 degrees, and
is positive if their angle is less than 90 degrees.
If we multiply the (component of) force F⋅T by distance, we get work.
Consequently, we can think of F⋅T as a scalar-valued function that gives work per
unit length along the slinky. To get the total work, we simply need to take the line
integral of this scalar-valued function7. In other words, the total work is
where C is the path of the slinky. This integral adds up the product of force (F⋅T)
and distance (ds) along the slinky, which is work.
To derive a formula for this work, we use the formula for the line integral of a
scalar-valued function7 f in terms of the parameterization c(t),
When we replace f with F⋅T, we get
where for our parameterization c(t) of the slinky, a=0 and b=6π.
We can simplify this further. Remember that T(t)=c′(t)/∥c′(t)∥. Consequently, the
two occurrences of ∥c′(t)∥ cancel, and we are left with
We usually write Tds as ds. Hence, the important formula to know is that the line
integral of a vector field is
where the curve C is parameterized by the function c(t) for a≤t≤b.
Source URL: http://mathinsight.org/line_integral_vector_field_introduction
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
www.saylor.org
Page 3 of 5
The fundamental role of line integrals in vector calculus
The line integral of a vector field plays a crucial role in vector calculus. Out of
the four fundamental theorems of vector calculus8, three of them involve line
integrals of vector fields. Green's theorem9 and Stokes' theorem10 relate line
integrals around closed curves11 to double integrals12 or surface integrals13. If
you have a conservative vector field14, you can relate the line integral15 over a
curve to quantities just at the curve's two boundary points. It's worth the effort to
develop a good understanding of line integrals.
Source URL: http://mathinsight.org/line_integral_vector_field_introduction
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
www.saylor.org
Page 4 of 5
Notes and Links:
1. http://mathinsight.org/line_integral_scalar_function_introduction
2. http://mathinsight.org/vector_field_overview
3. http://mathinsight.org/parametrized_curve_arc_length
4. http://mathinsight.org/parametrized_curve_introduction
5. http://mathinsight.org/parametrized_curve_derivative_location_velocity
6. http://mathinsight.org/unit_vector_definition
7. http://mathinsight.org/line_integral_scalar_function_introduction
8. http://mathinsight.org/fundamental_theorems_vector_calculus_summary
9. http://mathinsight.org/greens_theorem_idea
10. http://mathinsight.org/stokes_theorem_idea
11. http://mathinsight.org/line_integral_circulation
12. http://mathinsight.org/double_integral_introduction
13. http://mathinsight.org/surface_integral_vector_field_introduction
14. http://mathinsight.org/conservative_vector_field_introduction
15. http://mathinsight.org/gradient_theorem_line_integrals
Source URL: http://mathinsight.org/line_integral_vector_field_introduction
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
www.saylor.org
Page 5 of 5