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