Line integral of a scalar field:
Assume is a smooth function,
where is the domain of .
where is a smooth curve.
Line integral of a vector field:
Assume is a smooth vector function,
where is a smooth curve.
Assume is a vector field,
where is the domain and denotes the boundary of The orientation of satisfies the left hand rule. That means if you walk along the boundary of , the domain must be on your left.
is a simply connected region with boundary consisting four boundaries , the orientation is counterclockwise.
In the first graph, which denotes the boundary of has only one closed curve and the orientation of is counterclockwise. However, in the second graph, contains two curves, i.e. the blue one and the red one. The orientation on the blue one which is the outer boundary of is counterclockwise, the orientation on the red one which in the inner boundary of is clockwise. That means if you walk along the boundary of , the domain must be on your left. This is the left hand rule.
Corollary of Green’s Theorem
Assume is a domain in the plane, denotes the area of , then the area can be calculated from the following formulas:
Fundamental Theorem of Line Integral:
where denotes a curve from initial point to terminal point and is a scalar field.
Conservative Vector Field:
1. is called a conservative vector field, if there exists a scalar field such that . It is equivalent to these conditions:
2. is called a conservative vector field, if there exists a scalar field such that . It is equivalent the condition
A key property of a conservative vector field is that its integral along a path depends only on the endpoints of that path, not the particular route taken.
For example, if or is a conservative vector field, then the value of the line integral depends only on the initial point and terminal point of the curve . That means if is a conservative vector field, the curves and have the same initial points and terminal points, then these two line integrals are equal: . For this reason, a line integral of a conservative vector field is called path independent.
Question 1. For each non-zero constant , let denote the curve , where Let
Find the minimum value of in the domain
Method (i). Use the definition of line integration.
Since , , ,
The minimum value is taken at the
Method (ii). Use Green’s Formula.
Consider the domain bounded by and and
From Green’s Formula, pay attention to the orientation,
The derivative of is the minimum value is taken at and
Question 2. Prove the area of the disc with radius R is .
Method (i). Definition of Integration.
Method (ii). Green Formula
For on where
Question 3. Prove the area of the ellipse is .
Solution. It is similar to Question 2, and
Question 4. Calculate
where consists two line segments: from (0,0,0) to (1,0,2), and from (1,0,2) to (3,4,1).
Method (i). From the definition of line integral,
Method (ii). Check the vector field is a conservative vector field. Since
Therefore, is a conservative vector field, and we can assume , i.e. Hence,
for some constant
Therefore, the answer is
Question 5. Evaluate
where C denotes the boundary with positive orientation of the region between the circles and with
Method (i). The definition of the line integral.
On circle it is counterclockwise, is from 0 to
On circle it is clockwise, is from to 0.
On the circle ,
Pay attention to the orientation, we get the answer is
Method (ii). Green’s Theorem.