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One way to write the Fundamental Theorem of Calculus is:
$$\int_a^b f'(x)\,dx = f(b)-f(a).\]
That is, to compute the integral of a derivative \(f'\) we need only compute the values of \(f\) at the endpoints. Something similar is true for line integrals of a certain form.
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.
Proof
We write \({\bf r}=\langle x(t),y(t),z(t)\rangle\), so that \({\bf r}'=\langle x'(t),y'(t),z'(t)\rangle\). Also, we know that \(\nabla f=\langle f_x,f_y,f_z\rangle\). Then
$$\int_C \nabla f\cdot d{\bf r} = \int_a^b \langle f_x,f_y,f_z\rangle\cdot\langle x'(t),y'(t),z'(t)\rangle \,dt=\int_a^b f_x x'+f_y y'+f_z z' \,dt.\]
By the chain rule (see section 14.4) \(f_x x'+f_y y'+f_z z'=df/dt\), where \(f\) in this context means \(f(x(t),y(t),z(t))\), a function of \(t\). In other words, all we have is
$$\int_a^b f'(t)\,dt=f(b)-f(a).\]
In this context, \(f(a)=f(x(a),y(a),z(a))\). Since \({\bf a}={\bf r}(a)=\langle x(a),y(a),z(a)\rangle\), we can write \(f(a)=f({\bf a})\)---this is a bit of a cheat, since we are simultaneously using \(f\) to mean \(f(t)\) and \(f(x,y,z)\), and since \(f(x(a),y(a),z(a))\) is not technically the same as \(f(\langle x(a),y(a),z(a)\rangle)\), but the concepts are clear and the different uses are compatible. Doing the same for \(b\), we get
$$\int_C \nabla f\cdot d{\bf r} = \int_a^b f'(t)\,dt=f(b)-f(a)=f({\bf b})-f({\bf a}).\]
\( \square \)
This theorem, like the Fundamental Theorem of Calculus, says roughly that if we integrate a "derivative-like function'' (\(f'\) or \(\nabla f\)) the result depends only on the values of the original function (\(f\)) at the endpoints.
If a vector field \(\bf F\) is the gradient of a function,
\[{\bf F}=\nabla f\]
then we say that \(\bf F\) is a conservative vector field. If \(\bf F\) is a conservative force field, then the integral for work, \(\int_C {\bf F}\cdot d{\bf r}\), is in the form required by the Fundamental Theorem of Line Integrals. This means that in a conservative force field, the amount of work required to move an object from point \(\bf a\) to point \(\bf b\) depends only on those points, not on the path taken between them. In physics, forces that can ascribed to a conservative vector field are called conservative forces and are important for many applications.
Example \(\PageIndex{2}\):
An object moves in the force field
\[{\bf F}=\left \langle {-x\over (x^2+y^2+z^2)^{3/2}},{-y\over (x^2+y^2+z^2)^{3/2}},{-z\over (x^2+y^2+z^2)^{3/2}}\right\rangle,\]
along the curve \({\bf r}=\langle 1+t,t^3,t\cos(\pi t)\rangle\) as \(t\) ranges from 0 to 1. Find the work done by the force on the object.
Solution
The straightforward way to do this involves substituting the components of \({\bf r}\) into \(\bf F\), forming the dot product \({\bf F}\cdot{\bf r}'\), and then trying to compute the integral, but this integral is extraordinarily messy, perhaps impossible to compute. But since \({\bf F}=\nabla (1/\sqrt{x^2+y^2+z^2})\) we need only substitute:
\[\int_C {\bf F}\cdot d{\bf r}= \left.{1\over \sqrt{x^2+y^2+z^2}}\right|_{(1,0,0)}^{(2,1,-1)}={1\over\sqrt6}-1.\]
Another immediate consequence of the Fundamental Theorem involves closed paths. A path \(C\) is closed if it forms a loop, so that traveling over the \(C\) curve brings you back to the starting point. If \(C\) is a closed path, we can integrate around it starting at any point \(\bf a\); since the starting and ending points are the same,
$$\int_C \nabla f\cdot d{\bf r}=f({\bf a})-f({\bf a})=0.\]
For example, in a gravitational field (an inverse square law field) the amount of work required to move an object around a closed path is zero. Of course, it's only the net amount of work that is zero. It may well take a great deal of work to get from point \(\bf a\) to point \(\bf b\), but then the return trip will "produce'' work. For example, it takes work to pump water from a lower to a higher elevation, but if you then let gravity pull the water back down, you can recover work by running a water wheel or generator. (In the real world you won't recover all the work because of various losses along the way.)
To make use of the Fundamental Theorem of Line Integrals, we need to be able to spot conservative vector fields \(\bf F\) and to compute \(f\) so that \({\bf F}=\nabla f\). Suppose that \({\bf F}=\langle P,Q\rangle = \nabla f\). Then \(P=f_x\) and \(Q=f_y\), and provided that \(f\) is sufficiently nice, we know from Clairaut's Theorem that \(P_y=f_{xy}=f_{yx}=Q_x\). If we compute \(P_y\) and \(Q_x\) and find that they are not equal, then \(\bf F\) is not conservative. If \(P_y=Q_x\), then, again provided that \(\bf F\) is sufficiently nice, we can be assured that \(\bf F\) is conservative. Ultimately, what's important is that we be able to find \(f\); as this amounts to finding anti-derivatives, we may not always succeed.
Example \(\PageIndex{3}\)
Find an \(f\) so that \(\langle 3+2xy,x^2-3y^2\rangle = \nabla f\).
Solution
First, note that
\[{\partial\over\partial y}(3+2xy)=2x\qquad\hbox{and}\qquad{\partial\over\partial x}(x^2-3y ^2)=2x,\]
so the desired \(f\) does exist. This means that \(f_x=3+2xy\), so that \(f=3x+x^2y+g(y)\); the first two terms are needed to get \(3+2xy\), and the \(g(y)\) could be any function of \(y\), as it would disappear upon taking a derivative with respect to \(x\). Likewise, since \(f_y=x^2-3y^2\), \(f=x^2y-y^3+h(x)\). The question now becomes, is it possible to find \(g(y)\) and \(h(x)\) so that
\[3x+x^2y+g(y)=x^2y-y^3+h(x),\]
and of course the answer is yes: \(g(y)=-y^3\), \(h(x)=3x\). Thus, \(f=3x+x^2y-y^3\).
We can test a vector field \({\bf F}=\langle P,Q,R \rangle\) in a similar way. Suppose that \( \langle {P,Q,R} \rangle =\langle {f_x,f_y,f_z} \rangle \). If we temporarily hold \(z\) constant, then \(f(x,y,z)\) is a function of \(x\) and \(y\), and by Clairaut's Theorem \(P_y=f_{xy}=f_{yx}=Q_x\). Likewise, holding \(y\) constant implies \(P_z=f_{xz}=f_{zx}=R_x\), and with \(x\) constant we get \(Q_z=f_{yz}=f_{zy}=R_y\). Conversely, if we find that \(P_y=Q_x\), \(P_z=R_x\), and \(Q_z=R_y\) then \(\bf F\) is conservative.
Contributors
Integrated by Justin Marshall.
