Section 6.2 The Indefinite Integral
A major goal of the first part of this course was the extention of the derivative to multivariable functions: particularly to parametric curves and scalar fields. Now, the course pivots to pursue the matching goal of extending the integral to multivariable functions. I will consider new kinds of integrals involving parametric curves in
Section 10.2, as part of the material called Vector Calculus. For the start, though, I’m still interesting in scalar fields. What should it mean to integrate a scalar field? Well, there are two kinds of integral: definite and indefinite, so the question needs to asked twice. First, I’m going to think about indefinite integrals.
The indefinite integral
\(\int f dx\) was a notation for
all anti-derivatives of the single-variable function. Is there a multivariable version of this anti-derivative?
Chapter 4 tried to holistically extend the derivative to multi-variable functions. The result was a collection of construction: partial derivatives, tangent (hyper)planes, gradients and the Jacobian matrix. However, none of these notions of the derivative gave a new multi-variable funciton. Therefore, there isn’t a nice notion of anti-derivative: going back doesn’t make sense, since going forward led to more complicated constructions. If
\(f(x,y,z)\) is a scalar field, then
\(\int
f\) doesn’t have any clear meaning.
I can, however, take indefinite integrals in one variable, much like partial derivatives. If
\(f(x,y,z)\) is a continuous scalar field, then the following three integrals are the counterparts of partial differentiation. In each of the following, I pretend the other two variables are constant; only the variable indicated by the infintesimal term is non-constant.
\begin{align*}
\int f(x,y,z) dx \amp\\
\int f(x,y,z) dy \amp\\
\int f(x,y,z) dz \amp
\end{align*}
In addition to extending the integral, a major theme of this course will be extensions of the fundamental theorem. Let me remind you of the various forms of the fundamental theorem.
Theorem 6.2.1.
Let
\(f: [a,b] \rightarrow \RR\) be a continuous function. Then assume
\(F(x)\) is any antiderivative of
\(f\text{,}\) i.e.
\(F^\prime(x) = f(x)\text{.}\) Any of the following expressions are versions of the Fundamental Theorem of Calculus.
\begin{gather*}
\int_a^b f(x) dx = F(b) - F(a)\\
\int_a^b F^\prime(x) dx = F(b) - F(a)\\
\int f(x) dx = F(x) + c \\
\int F^\prime(x) dx = F(x) + c
\end{gather*}
All these equations essentially encode the same concept: that integration and differentiation are inverse processes.
The fundamental theorem deserves its name: it really is the basis for single variable calculus, giving the deep and remarkable connection between differentiation and integration. As this course extends the derivative and the integral to multivariable situations, there will be new opportunities to create connections between new derivatives and new integrals in the spirit of the fundamental theorem. A major goal of this course is to make new theorems that extend the fundanmental theorem. Therefore, whenever I can, I will be looking for these fundamental-theorem type constructions.
Here, I had defined partial derivatives and indefinite integrals in one variable. These are essentially one-variable constructions; I am just pretending all the other variables are constant. Therefore, the conventional single-variable fundamental theorem applies. Here is how I would write the fundamental theorem for a function of three varialbes, using partial derivatives and indefinite integrals in one variable.
\begin{align*}
\int \frac{\del}{\del x} f(x,y,z) dx \amp = f(x,y,z) + g(y,z)\\
\int \frac{\del}{\del y} f(x,y,z) dy \amp = f(x,y,z) + g(x,z)\\
\int \frac{\del}{\del z} f(x,y,z) dz \amp = f(x,y,z) + g(x,y)
\end{align*}
Note that the ‘constants of integration’
\(g\) now may involve the other variables. When I do single variable integration or partial differentiation, I pretend all the other variables are constant; therefore, the constant term can include those variables.
There is nothing more to be said about the indefinite integral and this is as far as I can extend the notation. All the future extensions of the integral (and there are many!) will be extensions of the definite integral.
