   Chapter 16, Problem 4RCC

Chapter
Section
Textbook Problem

(a) Define the line integral of a vector field F along a smooth curve C given by a vector function r(t).(b) If F is a force field, what does this line integral represent?(c) If F = ⟨P, Q, R⟩, what is the connection between the line integral of F and the line integrals of the component functions P, Q, and R?

(a)

To determine

To define: The line integral of vector field F along smooth curve C.

Explanation

The integrals which are done over a curve instead of a interval are referred as line integrals.

Consider a smooth curve C, atb given by vector function r(t)=x(t)i+y(t)j+z(t)k with parametric equations,

x=x(t)y=y(t)z=z(t)

The first derivative of vector function r(t)=x(t)i+y(t)j+z(t)k (r(t)) is continuous, and not equal to zero, r(t)0 as curve C is a smooth curve. Divide the parametric interval of curve C which is [a,b] , into the equal width of n subintervals as [ti1,ti] and subarcs of curve C as Δs1,Δs2,...,Δsn .

Hence, the parametric equations at simple point Pi(xi,yi,zi) for ith subarc are,

xi=x(ti)yi=y(ti)z=z(ti)

Consider a vector field F which includes the domain of curve C. Then the line integral of function is equal to the sum of product of value of f at point (xi,yi,zi) and subarc length Δsi .

The line integral of vector field F over smooth curve C is,

CFdr=limni=1nf(xi,yi,zi)Δsi (1)

Consider the length of subarc Δsi as L.

Re-modify the equation (1).

CFdr=f(x(t),y(t),z(t))L (2)

Write the expression for length of C on 3 (L) .

L=ab(dxdt)2+(dydt)2++(dzdt)2dt

Substitute ab(dxdt)2+(dydt)2+(dzdt)2dt for L in equation (2),

CFdr=F(x(t),y(t),z(t))(ab(dxdt)2+(dydt)2+(dzdt)2dt)

CFdr=abF(r(t))(dxdt)2+(dydt)2+(dzdt)2dt{F(r(t))=F(x(t),y(t),z(t))} (3)

Write the expression for CFdr .

F(r(t))=F(r(t))T(t) (4)

Here,

T(t) is a unit tangent vector at point (x,y,z) .

Write the expression for unit tangent vector T(t) .

T(t)=r(t)|r(t)|

Here,

|r(t)| is magnitude of first derivative of vector function r(t)

(b)

To determine

To explain: If the vector field F is force, then what does the line integral represent.

(c)

To determine

To explain: The relation between the line integral of F and line integrals of component functions P, Q, and R.

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