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In this section we will give the fundamental theorem of calculus for line integrals of vector fields. This will illustrate that certain kinds of line integrals can be very quickly computed. We will also give quite a few definitions and facts that will be useful.
Theorem: Fundamental Theorem of Line Integrals. Suppose a curve \(C\) is given by the vector function \({\bf r}(t)\), with \({\bf a}={\bf r}(a)\) and \({\bf b}={\bf r}(b)\). Then $$\int_C \nabla f\cdot d{\bf r} = f({\bf b})-f({\bf a}),\] provided that \(\bf r\) is sufficiently nice.
The Fundamental Theorem of Line Integrals is a precise analogue of this for multi-variable functions. The primary change is that gradient rf takes the place of the derivative f 0 in the original theorem.
Fundamental Theorem for Line Integrals – In this section we will give the fundamental theorem of calculus for line integrals of vector fields. This will illustrate that certain kinds of line integrals can be very quickly computed.
Fundamental theorem of line integrals: If F~= rf, then Z b a F~(~r(t)) ~r0(t) dt= f(~r(b)) f(~r(a)) : The proof of the fundamental theorem uses the chain rule in the second equality and the funda-mental theorem of calculus in the third equality of the following identities: Z b a F~(~r(t)) ~r0(t) dt= Z b a rf(~r(t)) ~r0(t) dt= Z b a d dt
Understand the fundamental theorem of line integrals, verify path independence for conservative vector fields and find a potential functions.
The following theorem says that we can evaluate the line integral of a conservative vector field, that is, the gradient field of a potential functionf, simply by knowing the value of fat the endpoints of C.
In this section we are going to evaluate line integrals of vector fields. We’ll start with the vector field, \[\vec F\left( {x,y,z} \right) = P\left( {x,y,z} \right)\vec i + Q\left( {x,y,z} \right)\vec j + R\left( {x,y,z} \right)\vec k\]
Significance of the fundamental theorem. For gradient fields F the work integral F · dr depends only on the endpoints of the path. C. We call such a line integral path independent. The special case of this for closed curves C gives: F = vf ⇒ F · dr = 0 (proof below). C. x C ,0) (1, 2) 1
The theorem The fundamental theorem of line integrals is: If F~= rfis a gradient eld then R b a F~(~r(t)) ~r0(t) dt= f(~r(b)) f(~r(a)). Proof: chain rule: R b a rf(~r(t))~r0(t) dt= R b a d dt f(~r(t)) dt= f(~r(b)) f(~r(a)). The last step is due to the fundamental theorem of calculus. Figure 1. We see the gradient eld F~(x;y;z) = hy;x;zi. It has ...