Line Integrals

Line integrals calculate summations of some quantifiable object X (for instance, density) over a path. We discussed in class how line integrals can also apply to paths that turn and go around in space. To calculate such integrals, we simply need to describe the path as a piece-wise function and integrate accordingly.

In one of our problems, we reversed the path our particle in our line integral would take and we got the exact opposite answer. But what if instead of reversing the path, we reversed the vector field? The answer we arrived at was that it would not yield the same answer. The reversed vector field wouldn't affect the path that the particle took in the same way.

If you look closely, the angle between the original vector field and the tangent to the curve is slightly larger than that of the reversed vector field. So reversing a vector field won't necessarily generate the opposite amount of net work done by the particle, but does yield some interesting phenomenon in physics.

For instance, if you shatter a glass, every resulting shard has a force vector pointing outwards from the origin of, shall we say, collapse? Well, what if we reversed all those vectors? Would the glass come back together? (Link below)

Skip to around 6 seconds and imagine freezing a glass that had just shattered and reversing all those force vectors. And perhaps this would happen.

It's definitely an interesting thing to picture in your head!

Vector Fields - Conservative Fields in Space

As it turns out, determining whether or not a vector field is conservative in space is radically different than that of a vector field in the plane. To determine whether or not a vector field is conservative in space (and by this, I mean 3D vector fields), we need to take the cross product of the gradient and our vector function. This yields a matrix:

This is called the curl of vector field F. If the curl is the zero vector, then the vector field is conservative.

Vector Fields - Conservative Fields in Planes

Recall that from the typical functions that we are familiar with, we have "continuous" and "discontinuous" functions. The well-behaved functions are called "continuous." In the case of vector fields, well-behaved fields are termed conservative.

A vector field is mathematically defined as conservative if and only if it exists as the gradient of another function, called the potential function. In other words:

But this doesn't mean that given some vector field, you absolutely must engage in complicated mathematics to determine whether or not it supports abortion. Take for instance:

We know that if this is the gradient of some function, it is conservative. The "i" component of the vector is the partial derivative with respect to x; likewise, the "j" component is the partial derivative with respect to y. Easily enough, we can see that the following function indeed works:

But what if we wanted a more reliable method of confirming this? The following would do just that. Consider this general vector formula:

That vector is conservative if:

And why does this make sense? If the vector field "F" is the gradient of some potential function, then it should be in this format:

If this function is "well-behaved", then its mixed partials should be equal. Thus, we arrive at our conservative formula.

Now, this process gets radically different in three-dimensions, but it isn't all that bad. But lets leave that for another post and another time.

Happy vector conserving!

Vector Fields - Introduction & Basics

We continued our discussion on vector fields in class today. Essentially, a vector field is simply another function, which we are familiar with.


Rectangular functions of two variables takes an x-coordinate as an input and outputs a y-coordinate (or vice versa). Three variable rectangular functions take a point in the plane (both x and y) and outputs a point in space (associating a z-coordinate). Vector valued functions take x-values and output vectors.

Finally, vector fields take in points (both x and y), and output vectors.

Keep in mind that vector fields cover the entire domain of the plane, or space, depending on how many dimensions you are working with. In class, we graphed and analyzed some basic vector fields, associating them with some natural phenomena.

1.

The most basic one: v (x,y) = <x , y> is described as consisting of vectors pointing radially outwards with magnitudes proportional to its distance from the origin. We associated it with the "Big Bang," as the universe's expansion is supposedly accelerating.

2.


The reverse of this vector field: v (x,y) = <-x , -y> essentially reverses every vector in the field. We described the resulting field as the opposite of the Big Bang, or the "Big Crunch."

3. 


Our third basic vector field looks a little more natural. We associated the field described by: v (x,y) = <y , x> with two water currents colliding with each other. Note that the reverse vector: v (x,y) = <-y , -x> would just have the "entrance" and "exit" of the vectors reversed.

4.


And finally, our favorite of the day, the "Toilet Flush": v (x,y) = <y , -x>. The reverse of this function: v (x,y) = <-y , x> would simply be a toilet being flushed on the other side of the world.

The drawings I added to go with the vector fields aren't entirely correct, but it's just there to give a less cluttered visualization.

To draw your own vector fields, just plot a few vectors at various points in the plane and try to see a pattern. They're usually symmetric, whether it's about the origin in the case of the "toilet flush" or about the lines y = x and y = -x in the case of "water currents".

Happy...vector field-ing?

Multiple Integration - Spherical Coordinates

Sorry for the late post (haven't gotten around to updating the blog over spring break)!

But yes, it's another coordinate system. Just as with cylindrical coordinates mentioned in the last post, spherical is somewhat of a cousin of our old polar coordinates.

(Refer to the blog archive for a refresher on polar coordinates.)

There are three components to coordinatizing points in spherical:

Both "rho" and "theta" are exactly the same as polar and cylindrical. The only difference between spherical and cylindrical, however, is the final component. This difference makes sense, though. Take a moment and imagine a cylinder. It is, in essence, a circle extruded infinitely in the z-direction. Therefore, it only makes sense for it to have a z-component to its point-coordinates. For a sphere, however, you have "phi", the extent of the surface from the positive z-axis.

Imagine a circle on the yz-plane extruded down to the xy-plane as depicted in the image below. If the top of the circle is only brought down to the xy-plane, it has only completed 1/4 of its way around the coordinate plane, so "phi" is pi/4. Keep in mind, however, that since "phi" is only measured from the positive z-axis, the highest possible value for "phi" is pi. This prevents doubling-over when measuring "phi".

To integrate, simply keep the following conversions from rectangular to spherical in mind and integrate accordingly:

Also, just as with cylindrical coordinates, there is a new "dV" in spherical, which may seem a little daunting at first sight, but isn't that bad once the trigonometric functions eliminate the angles involved with "theta" and "phi":

A short sample:
Consider the following sphere of radius 4 (it looks like a cracked egg, but bear with me):

Let's write the triple integral that will result in it's enclosed volume.

• It extends from the positive z-axis to the negative z-axis, so "phi" is pi.
• It extends all the way around the xy-plane, so "theta" is two-pi.
• And it's radius is 4, so "rho" is 4.

And our integral should look something like this:

It may seem a little confusing at first, but with a little practice, it'll get easier.
Happy spherical integrating!

Multiple Integration - Cylindrical Coordinates

Having new coordinate systems to work with may seem at first like an excess of information, but in some cases in mathematics, having another coordinate system to work with can actually make a difficult problem much easier.

Imagine cylindrical coordinates as the polar plane extruded along the z-axis. (Infinitely up and down). By this logic, the three components of cylindrical coordinates are:

This z-component makes all the difference. In polar coordinates, r=3 yields a circle with radius 3. In cylindrical, you get exactly what its name implies—a cylinder of radius 3. It's the circle we had in polar extruded infinitely in the z-direction.

But how do we integrate in cylindrical? Similar to how when we converted our dA or dV in rectangular integration to polar, we get an r with our differentials, in cylindrical integration, we get:

Simply express "theta" and the radius (r) as you would in polar. But now, you simply include the z-range and integrate the function accordingly.


To convert between rectangular and cylindrical, refer to the conversions mentioned in the March post: "Multiple Integration - Double w/ Polar".


Happy integrating in cylindrical!

Center of Mass - Double and Triple Integration

All this integrating can be tied together in center of mass. Imagine you have a cube that's gets denser as you move further out towards its corners. Where would it's "center of mass" or "core" be? That can be easily accomplished using multiple integration!

• If it's a 2D surface, use a double integral.
• If it's a 3D volume, use a triple integral.

First, lets describe the mass of a volume. In general, the formula for the mass of a given solid is:

To describe "x ranges" and "y ranges" for centers of mass, these ranges are described as "moments"--how the object acts in a certain direction, literally, at a certain moment in time. The general formulas for these are different for 2D and 3D objects. But the rules are somewhat similar.

Notice that the main difference here is that instead of just having the surface vary along two axes, in 3D space, the object can vary along planes. Also, notice how within the integral, the function of density p(x,y,z) is being multiplied by an additional variable (either x, y, or z). Conveniently, by dividing these moments by the mass, you get one component of the center of mass. Put the components together and you'll get the coordinate of the center of mass for that particular surface or solid!

Happy integrating!

Multiple Integration - Triple

Triple integration may seem like you're just sticking another integral sign onto a double integral, but it comes with its own set of complications. Similar to how in double integration, you could reverse the order of integration (either dydx or dxdy), in triple integration, you have an array of six possible orders to integrate. For this reason, it is imperative to sketch surfaces properly and keep your limits of integration organized. For instance, suppose you have the plane x + y + z = 1 in the first octant.

Lets describe the volume bounded by the plane in the first octant in the order dzdydx. This essentially means describe how x varies, then y, then z.

• Well, the range for x is easy: from 0 to 1.
• While x ranges from 0 to 1, y ranges from 0 to the line y=x, so from 0 to x.
• And while y ranges from 0 to x, z ranges from 0 to the plane, so from 0 to 1-x-y.

Thus, our triple integral looks something like:

Now what if we switched the order of integration to dxdydz? Well, now we look at z first, then y, then x last.

• Z ranges from 0 to 1.
• This is where the change comes in. As z varies, y doesn't go out to the plane because our "depth" here is the x-coordinate and we haven't described that yet. To determine the limits on the y-axis, we have to look at the trace of the plane.

• Therefore, y ranges from 0 to 1 - z.
• And x ranges from 0 to the plane, so from 0 to 1 - y - z.

Thus, our triple integral in this new order is:

Remember that order matters!

Happy triple integrating!