PHY 101: Vector Integration

What is Integration?

Integration can be thought of as a continuous analogue of sum (\(\sum_{ }^{ }\)). We integrate or simply add infinitesimal small quantities together to form a continuous chain. Integration can be of 3 kinds rather I would call it 3 ways of integration. Linear, Area and Volume way. To show what each of them looks like here is a visual representation:

$$\begin{aligned}& Linear \hspace{1mm}integration:\int_{ }^{ }f(x)dx\\ \\ &Area\hspace{1mm}or\hspace{1mm} surface\hspace{1mm} integration: \iint_S f(x,y)dxdy \\ \\ & Volume \hspace{1mm} integration: \iiint_V f(x,y,z)dxdydz \end{aligned}$$

You might have already noticed that the number of integration symbols (\(\int_{ }^{ }\)) increases with the increase in the number of variables. Hence, most books adopt the notation of calling these single, double and triple integrations. We at physics are creatures of simplicity and thus have kept it easy to remember.

Let us talk about each in some detail!

Single or Linear Integration

In general, any integral which is to be evaluated along a curve is a line integral. Think of it as what would happen to a vector function when we calculate it along a curve or a line, one of the examples that comes to mind is Work done by any object. Remember the grade 12 equation of Work done being \(F.dr\) think of it if the force acts on an object along a curve C, then work done is simply the magnitude of the force along that curve.

We can now say that the force acting along a curve from point x1 to x2 will give us the work done by that force, \(\int_{x1}^{x2} \vec{F}.dx\)

if the integration along the curve starts and ends at the same point we call it a closed-loop integral (\(\oint\)). you might be tempted to say this will be zero but that is not the case. To know more click here.

Note: The integration is independent of the curve taken joining the points x1 and x2. 

Double or Surface Integration

These can be thought of double integration of the function across two dimensions making up the surface. The integration is solved in the following order, first, we solve the inner integral and then the outer integral.

Similar to the previous case, we also have a closed surface integral when the surface chosen has no boundary. One of the ways to imagine is a cube or a cuboid, it is a closed surface with no boundary, not to be confused with edges which are boundaries to the face of the cube.

Triple or Volume Integration

These can be thought of triple integration of the function across three dimensions making up the volume. Similar to the double integral we solve the triple integral, first inner integral is solved, then further outward. often times we write the volume component i.e \(dxdydz =dV\) thus the integral becomes \(\iiint_V f(x,y,z)dV\)

I would advise you to solve plenty of problems on these topics to be comfortable with them.

The reason you need to be comfortable is actually so you can be confident when you change one of these integrals to the other. Yes, you read it correctly! we do convert these integrals from one form to the other. But for that, we must understand the various bounds in which these are calculated respectively.

We use three main formulas to achieve this purpose.

1. Gauss Divergence Theorem

2. Stokes' Theorem

3. Green's Theorem

let us discuss them one by one

Divergence Theorem of Gauss

$$\iiint_V \nabla \cdot \vec{A} dV=\iint_S \vec{A}.\vec{n} dS=\oint_S \vec{A}\cdot\vec{S}$$

It states that if V is the Volume bounded by a closed surface S and A is the vector function of the position with continuous derivatives. where n is the positive normal to the S.

Do not be fooled by the single closed integration, often times this is used in many books as the symbol denotes closed loop integrals but since it is acting on the surface S, it is a closed surface integral.

This is a very handy integral as you will see later. We can use this to evaluate any Double integral in 3D and serve as a cheeky shortcut.

Stokes' Theorem

$$\oint_C \vec{A}\cdot dr=\iint_S (\nabla \times \vec{A})\cdot n dS=\iint_S (\nabla \times \vec{A})\cdot d\vec{S}$$

It states that if S is an open, two-sided surface bounded by a closed non-intersecting curve, then if A has a continuous derivative where C is traversed in the positive direction, we can write the line integral in a double integral representation as shown above.

Stokes' theorem is particularly useful as it is the foundation upon which Green's theorem is based.

Green's Theorem

$$\oint_C Mdx+Ndy=\iint_S (\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y})dxdy$$

If S is the closed region in the xy plane bounded by a simple closed curve C and if M and N are continuous functions of x and y having continuous derivatives in S then the above expression holds true.

While knowing these definitions is useful we often tend to just use these expressions directly in Physics. Sometimes when solving a problem if we encounter a difficult integration we try to use the above three formulas to simplify the question and find a definitive answer.

Conclusion:

Here we are, having discussed the basics of vector calculus we can now move forward with Classical Mechanics. I hope you have understood the basics, and practice questions on these topics until you are comfortable with these symbols, as we move forward we'll encounter them a handful of times. Bare in mind I am not going to introduce the orthogonal system of coordinates now, we will discuss it some other day when we need it the most till then our examples and derivation will be based upon the cartesian system of coordinates only.

We'll meet again reader, until next time!