**Chapter 21**

August 21 2017

**Chapter 21 Line Integrals and Green's Theorem**
**21.1 Line Integrals in the Plane**
- this chapter brings together into a unified package several topics in multi variable calculus
- that are important for physical science and engineering
- which provides yet another way of extending ordinary integration to higher dimensions
- line integrals are used, for example
- to compute the work done by a variable force in moving a particle along a curved path from one point to another
- the main result of this chapter (Green's Theorem) uses partial derivatives to establish a connecting link between line integrals and double integrals
- this enables us to distinguish those vector fields that have potential energy functions from those that do not
- our first problem is to formulate a satisfactory concept of work
- if we push a particle along a straight path with a constant force F
- the work done by the force is the product of the component of F in the direction of motion and the distance the particle m oves
- W = F dot dR
- now suppose F is not constant
- but a vector function that varies from point to point throughout a certain region of the plane
- F = F(x, y) = M(x, y) i + N(x, y) j
- suppose the variable force pushes a particle along a smooth curve C in the plane
- where C has parametric equations
- x = x(t) and y = y(t)
- what is the work done by the force as the point of application moves along the curve
- the vector valued function is called a force field
- more generally
- a vector field in the plane
- is any vector valued function that associates a vector with each point in a certain plane region R
- in this context a function whose values are numbers is a scalar field
- for example f(x, y) = x2 y3 is a scalar field defined on the entire xy-plane
- every scalar field f(x, y) gives rise to the corresponding vector field
- del f(x, y) = grad f(x, y) = df/dx i + df/dy j
- called the gradient gradient field of f
- some vector fields are gradient fields
- most are not
- we shall see those vector fields that are also gradient fields are of special importance
- we now return to the problem of calculating the work done by the variable force
- F = M(x, y) i + N(x, y) j
- along the smooth curve C
- this leads to a new kind of integral called a line integral
- Sc F dot dR = Sc M(x, y) dx +_N(x, y) dy
- if Fk is the value of F at Pk
- the sum is E Fk dot dRk
- the line integral of F along C is
- Sc F dot dR = lim E Fk dot dRk
- the limit of the polygonal path are understood to approximate the curve C more and more closely
- the idea is F is almost constant along the short path segment dRk
- a quick intuitive way of constructing the line integral is
- if dR is the element of displacement along C
- then the corresponding work done by F
- is dW = F dot dR
- the total work is
- W = S dW = Sc F dot dR
- for additional insight into the meaning of this formula
- we think of the position vector R as a function of the arc length s measured from the initial point A
- since dR/ds is the unit tangent vector T
- Sc F dot dR = Sc F dot dR/ds ds = Sc F dot T ds
- so the line integral can be thought of as the integral of the tangent component of F along the curve C
- the line integral include ordinary integrals as special cases
- the parametric representation allow us to express everything in terms of t
- and to solve as an ordinary single integral with t as the variable of integration
- solving line integrals includes selecting the parametric representation used
- every curve C that we use with line integrals is understood to have a direction
- from its initial point to its final point
- even though the value of a line integral does not depend on the parament
- it does depend on the direction
- the line integral of a given vector field from one given point to another depends on the choice of the curve
- and has different values for different curves
**21.2 Independence of Path. Conservative Fields**
- if the vector field is the gradient of a scalar field
- for example
- f(x, y) = xy + y2
- del f = y i + (x + 2y) j = F
- recall from section 19.5 that in multi variable calculus the gradient plays a similar role to that of the derivative in single variable calculus
- the Fundamental Theorem is
- Sab f'(x) dx = f(b) - f(a)
- the corresponding result is
- Sc del f dot dR = f(B) - f(A)
- where f(x, y) is a scalar field and
- A and B are the initial and final points on C
- since the vector field is the gradient of the scalar field
- F = del f
- so the value of the line integral can be calculated irregardless of the path the curve takes
- Sc F dot dR = Sc del f dot dR
- this is the Fundamental Theorem of Calculus for Line Integrals
- if a vector field F is the gradient of some scalar field f in a region R
- so F = del f in R
- and if C is any piecewise smooth curve on R with initial and final points A and B
- then
- Sc F dot dR = f(B) - f(A)
- the Fundamental Theorem can sometimes be used in the practical task of evaluating line integrals
- nevertheless
- its main importance is theoretical
- the line integral of a gradient field is independent of the path
- if B = A
- then Sc F dot dR for every closed path C
- any of these three conditions is equivalent to the others
- the force field F is called conservative if it is the gradient of a scalar field
**21.3 Green's Theorem**
- Green's Theorem establishes an important link between line integrals and double integrals
- consider a vector field
- F = M(x, y) i + N(x, y) j
- defined on a certain region in the xy-plane
- we take up the question of whether the condition
- dM/dy = dN/dx
- is sufficient to guarantee F is conservative
- that is
- that F is the gradient of some scalar field f
- Sc M dx + N dy = SS R [dN/dx - dM/dy] dA
- this is Green's Theorem
- that a line integral around a closed curve equals a certain double integral over the region inside the curve
- if C is a piecewise smooth, simple closed curve that bounds a region R
- and if M(x, y) and N(x, y) are continuous and have continuous partial derivatives along C and throughout R
- then the equation holds
- if the domain of definition of the vector field F = M(x, y) i + N(x, y) j is imply connected
- then F is conservative if and only if the condition
- dM/dy = dN/dx is satisfied
- if a given vector field F is known to be conservative
- so F = del f for some function f(x, y)
- how do we find f
- such a function is called the potential function of F
- by integrating
- df/dx = M, df/dy = N