MULTIVARIABLE CALCULUS 12

Integral Theorems

The major integral theorems connect local derivative information inside a region with accumulated behaviour on its boundary.

The main idea

Earlier calculus connected derivatives and integrals through the Fundamental Theorem of Calculus.

In multivariable calculus, similar ideas connect:

\[ \text{gradient}, \qquad \text{curl}, \qquad \text{divergence} \]

with integrals over curves and surfaces.

\[ \boxed{ \text{local derivative information} \longleftrightarrow \text{boundary accumulation} } \]

1. The Fundamental Theorem of Calculus

In one variable,

\[ \boxed{ \int_a^b F'(x)\,dx = F(b)-F(a) } \]

The integral of a derivative over an interval depends only on the boundary points.

This is the pattern that the higher-dimensional theorems extend.

2. Fundamental theorem for line integrals

Suppose

\[ \mathbf{F}=\nabla\phi. \]

Then

\[ \boxed{ \int_C \mathbf{F}\cdot d\mathbf{r} = \phi(B)-\phi(A) } \]

where \(A\) and \(B\) are the endpoints of the curve.

So integrating a gradient along a path depends only on the endpoints.

3. Why this resembles ordinary calculus

In one variable,

\[ F'(x) \]

is the derivative of a scalar function.

In several variables,

\[ \nabla\phi \]

plays the corresponding role.

\[ \boxed{ F' \longleftrightarrow \nabla\phi } \]

4. Path independence

\[ \mathbf{F}=\nabla\phi, \]

then the line integral between two points does not depend on the path.

Therefore,

\[ \boxed{ \text{gradient field} \Rightarrow \text{path independence} } \]

For a closed curve,

\[ \boxed{ \oint_C \mathbf{F}\cdot d\mathbf{r} = 0 } \]

5. Green's theorem

Let

\[ \mathbf{F} = \begin{pmatrix} P(x,y)\\ Q(x,y) \end{pmatrix}. \]

For a suitably smooth field and a positively oriented simple closed curve \(C\) enclosing a region \(R\),

\[ \boxed{ \oint_C P\,dx+Q\,dy = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \,dA } \]

This is Green's theorem.

6. What Green's theorem means

The left side measures circulation around the boundary.

The right side accumulates local rotational tendency over the interior.

\[ \boxed{ \text{circulation around boundary} = \text{total curl inside} } \]

7. Boundary and interior

Green's theorem connects two different kinds of objects:

\[ \text{one-dimensional boundary }C \]

and

\[ \text{two-dimensional region }R. \]

It converts a line integral around the boundary into a double integral over the interior.

8. Diagram

9. Stokes' theorem

Green's theorem extends to surfaces in three dimensions through Stokes' theorem.

Let \(S\) be an oriented surface with boundary curve \(C=\partial S\).

Then

\[ \boxed{ \oint_{\partial S} \mathbf{F}\cdot d\mathbf{r} = \iint_S (\nabla\times\mathbf{F}) \cdot\mathbf{n}\,dS } \]

10. Meaning of Stokes' theorem

The line integral around the boundary measures circulation.

The surface integral measures accumulated curl through the surface.

\[ \boxed{ \text{circulation around boundary} = \text{total rotational tendency through surface} } \]

11. Why Green's theorem is a special case

If the surface \(S\) lies flat in the \(xy\)-plane, then Stokes' theorem reduces to Green's theorem.

So Green's theorem can be viewed as the planar version of Stokes' theorem.

12. The divergence theorem

Let \(V\) be a three-dimensional region with closed boundary surface \(\partial V\).

Then

\[ \boxed{ \oiint_{\partial V} \mathbf{F}\cdot\mathbf{n}\,dS = \iiint_V \nabla\cdot\mathbf{F}\,dV } \]

This is the divergence theorem.

13. Meaning of the divergence theorem

The left side measures total outward flux through the boundary.

The right side accumulates divergence throughout the interior.

\[ \boxed{ \text{outward flux through boundary} = \text{total source strength inside} } \]

14. Why this makes sense

Imagine dividing a region into many tiny cells.

Flow leaving one cell may enter a neighbouring cell.

These internal contributions cancel when all cells are combined.

Only flow through the external boundary remains.

15. Diagram for the divergence theorem

16. The pattern behind all these theorems

The theorems may look different, but they share one structure:

\[ \boxed{ \text{integral of a derivative over a region} = \text{integral over its boundary} } \]

17. Dimension increases by one

The Fundamental Theorem of Calculus connects a one-dimensional interval with its zero-dimensional endpoints.

Green's theorem connects a two-dimensional region with its one-dimensional boundary.

The divergence theorem connects a three-dimensional volume with its two-dimensional boundary.

\[ \boxed{ \text{interior dimension} \rightarrow \text{boundary dimension one lower} } \]

18. Comparison of the main theorems

\[ \boxed{ \int_a^b F'(x)\,dx = F(b)-F(a) } \]

Fundamental Theorem of Calculus

\[ \boxed{ \int_C\nabla\phi\cdot d\mathbf{r} = \phi(B)-\phi(A) } \]

Fundamental Theorem for Line Integrals

\[ \boxed{ \oint_C \mathbf{F}\cdot d\mathbf{r} = \iint_S (\nabla\times\mathbf{F}) \cdot\mathbf{n}\,dS } \]

Stokes' Theorem

\[ \boxed{ \oiint_{\partial V} \mathbf{F}\cdot\mathbf{n}\,dS = \iiint_V \nabla\cdot\mathbf{F}\,dV } \]

Divergence Theorem

19. Local versus global information

Gradient, curl and divergence are local derivative quantities.

Line, surface and volume integrals collect information over larger regions.

\[ \boxed{ \text{derivatives} = \text{local behaviour} } \] \[ \boxed{ \text{integrals} = \text{accumulated behaviour} } \]

The integral theorems explain how the two are connected.

20. Connection with conservation laws

Suppose a density \(\rho\) and flux \(\mathbf{J}\) satisfy

\[ \frac{\partial\rho}{\partial t} + \nabla\cdot\mathbf{J} = 0. \]

Integrating over a region \(V\) gives

\[ \iiint_V \frac{\partial\rho}{\partial t} \,dV + \iiint_V \nabla\cdot\mathbf{J} \,dV = 0. \]

Using the divergence theorem,

\[ \iiint_V \frac{\partial\rho}{\partial t} \,dV + \oiint_{\partial V} \mathbf{J}\cdot\mathbf{n}\,dS = 0. \]

This says that change inside the region is balanced by flow through the boundary.

21. Why these theorems matter

The integral theorems are central in fluid mechanics, electromagnetism, continuum models, conservation laws and partial differential equations.

They allow us to move between local equations and global statements about entire regions.

22. The central idea

\[ \boxed{ \text{Fundamental Theorem} } \] \[ \downarrow \] \[ \boxed{ \text{Green} \rightarrow \text{Stokes} \rightarrow \text{Divergence} } \]

These are not isolated formulas. They are higher-dimensional versions of the same fundamental principle connecting differentiation and integration.