Lecture 11: Vector Calculus III 1. Key points Line integrals (curvilinear integrals) of scalar fields Line integrals (curvilinear integrals) of vector fields Surface integrals Maple int PathInt LineInt
2. Line integrals of scalar field The integral of a scalar field along a path is written as
where ds is infinitesimal segment along the path. There is another kind of the line integral of scalar fields
where dr is the element of the line along the path. In the Cartesian coordinates, it can be expressed as
Note that the result is a vector. This type of integrals is not common in physics. We discuss only the first kind. Example in physics The length of path: The time of travel:
For simplicity, we study only two dimensional cases in this lecture. Path given as y = y (x)
The line segment can be written as
Then, we have the line integral
Example
Consider a projectile motion
where a and b are x and y component
of the initial velocity, respectively. Find the length of the path.
=
=
Path given as a parametric function [x(s),y (s)] Using
and
, .
Then, we have the line integral Example 2.1 Consider a projectile motion
. Find the length of the path from t=0
to t=2b/g.
=
=
This answer is the same as the previous example as it should be.
Example 2.2 Evaluate the integral
along the path specified by
,
,
.
=
2 3
3. Line integrals for vector fields: Type 1 There are two types of line integrals or curvilinear integrals of a vector field along a path: . and .
where d is the element of line, an infinitesimal vector tangent to the path. In the Cartesian coordinates . The result of the first integral is scalar while the second one is vector. We discuss the first kind in this section. Expressing the vector field also in the Cartesian coordinates,
Then, the line integral is
The first integral in the right hand side the value of y and z must be specified. Similarly, in the second integral the value of x and y must be given. Hence, this integral does not have a unique value unless the relation among x, y and z is given. The relation does not have to be one-to-one relation. For simplicity, we study only two dimensional cases in this lecture. Examples in physics (1) Work (2) Scalar potential (3) Ampere theorem
Path consisting of multiple line segments. Consider a vector field . We integrate it along a path starting at (0,0) and ending at (1,1). We take a path consisting of two segments of straight lines: a horizontal segment from (0,0) to (1,0) and a vertical segment from (1,0) to (1,1). Than means when x is varied, y=0 and when y is varied, x=1. Then, we have =
= =2
1
y
0
1
x The path of integration, vector(s) tangent to the path, and vector-field arrows 0
(3.2.1)
Path given as a function y=y(x) Using we can eliminate y. . Example Consider a vector field (0,0) to (1,1). That means along y=x.
again. We integrate it along a straight line from
=
=
3 2
1
y
0
1
x The path of integration, vector(s) tangent to the path, and vector-field arrows
E x a mp le -3D
=
1 2
The path of integration, vector(s) tangent to the path, and vector-field arrows
Parametric path [x(t),y(t)] Using
and
, .
Example 1 Consider a vector field once more. This time, we integrate it along an arc of a circle centered at (1,0). The path is given by and .
(3.4.1.1) = (3.4.1.2)
1
y
0
1
x The path of integration, vector(s) tangent to the path, and vector-field arrows
Example 2 - Closed loop Integrate the vector field
along a circle of radius 1 centered at (0,0). ,
= =
1
y
0
1 x
The path of integration, vector(s) tangent to the path, and vector-field arrows
Line integral of Gradient Consider a line integral of gradient of a scalar field along a path starting from
to
.
Example: Work and potential energy Work is defined as
. If the force is conservative,
. Hence,
, where
and
are starting and ending points of the
path,respectively.
4. Line integrals for vector fields: Type 2 For the second kind of the line integral of a vector field is
. Since it is a cross
product of two vectors the result is a vector defined by
. Examples in physics Biot-Savart law: The magnetic field due to a steady current along a line is given by where
is the element of length along the line of the current and is
the position vector measured from the element of the line to the observation point. A loop of wire carrying a current is placed in a magnetic field . The force exerted on the wire by the magnetic field is given by .
Example
Consider a vector field
=
=
. Evaluate
for the circular path on the xy plane with radius . Using parametric representation of the path:
, (because
) (because
=0 (because
)
)
=0 (because
) =0 =0
Hence the integral is zero vector. This result is obvious because along the circle both product is zero.
and
are on the xy plane and their cross
5. Surface integrals The surface integral of scalar fields is written as , and the surface integral of vector fields as . where rule. atomic
is the surface element whose direction is normal to the surface determined by right-hand
In physics, the second form is by far the most common form of the surface integral. Examples in physics Gauss's law:
S).
.
Example Consider a unit cube centered at the origin. Its edges are parallel to the axes. Find S is all faces of the cube. There are six faces. Chose the outward direction is positive. For the face at
,
. .
Due to the cubic symmetry, all faces have the same integral value. Hence, .
where
Homework Homework 1 Consider a projectile motion
. Find the time of travel from x=0 to x=R.
Homework 2 1. Evaluate
.
2. Consider a force field . Calculate the work done by the force in moving from (1,1) to (3,3) along two different paths. (You pick two paths.) 3. Consider a force field . Calculate the work done by the force along a complete unit circle defined by
.
Homework 3 Evaluate
over the whole surface of the cylinder bounded by
.
