Vector & Tensor Fundamentals

RainmanTime

RainmanTime

Super Moderator
Greetings!

At the behest of Darby, a valued contributor to this forum, this thread is being started to help folks understand why vectors are a necessary mathematical construct when it comes to physics (more specifically, the dynamics of motion of a body). This should then lead to a natural discussion of how combinations of vectors lead to the mathematical necessity for the generalized concept of a tensor, and we will further understand how a vector is a special case of a generalized tensor. Furthermore, once we have the concepts of vectors solidly established (and answered any questions from those who may have them), I may also describe how the strict concept of a vector can be generalized in a manner that also applied to the dimensions of Mass and Time, in addition to their classical application to only spatial physics. Questions & contributions are certainly welcome. While I am used to being a teacher on such things, I don't claim to be the ultimate authority on all things vector and tensor.

IMO, I think the best place to begin is to establish the concept of a Scalar, and how it is different from a Vector, as this will certainly help us establish the basis for the vector itself (no pun intended!). A scalar corresponds to any generalized measurement which is not associated with any specific directional orientation in 3-D space. One good example of a Scalar measurement would be atmospheric temperature. If I look at the thermometer outside and see it is 65 degrees F outside, there is no depedency upon which direction the thermometer is facing. The temperature is 65 degrees whether it is facing N, S, E, W or some other off-nominal direction in between these. Another way of saying it would be to state that the air temperature has a Magnitude associated with it, but no specific Direction associated with it. In the classical formulation of physics, we would also say that the mass of a body is also a Scalar quantity, as we can denote the total Magnitude of the body's mass (Kg, or Slugs, or Pounds-mass), but there is no specific Direction associated with this measurement of mass. (In reality, we cannot and do not directly measure a body's mass, but rather we will measure its weight within the gravitational field of the earth, and then compensate for the gravitational field by dividing that weight by "g"). Another good example of a scalar measure would be the number of electrons associated with a specific element. It is just a number, with no spatial orientation necessary to understand that number (e.g. hydrogen has 1 electron). BTW: For further reference, we can refer to a Scalar quantity as a "tensor of Rank 0". It will be clear why we refer to it in this way as we progress. Just file that away in the "to be used later" bag of information.


Now is where we can finally introduce the concept of a Vector and how it goes beyond the concept of the Scalar. A Vector is defined as a physical measurement that possesses both Magnitude and Direction. IOW, to fully characterize this type of measurement it is not enough to just know the Magnitude. We must also understand how the Magnitude of this measurement is oriented in space. The most common example of a Vector quantity is the weight of an object, let's say the weight we read when we step on the scale. The scale will give us the scalar Magnitude (no pun intended, again!). However, the direction associated with any weight measurement that we make here on the surface of the earth is, by definition, associated with the Direction that gravity acts, namely towards the center of the earth. Therefore, my weight has both a scalar Magnitude of 170 pounds, and a vector Direction pointed downwards towards the center of the earth. Likewise, any measurement of force (of which weight is a specialized form of force measurement associated with a body's mass) will be a vector with both Magnitude and Direction. So when my car stalls on the road (which my Vette has never done, thank goodness) and I have to push it, I will exert a specific Magnitude of force against the car to get it rolling, and that force will be oriented in a specific Direction, namely in a horizontal direction parallel to the surface of the earth.

This is a good enough stopping point for Part 1, to allow folks to read/consider this material, and ask any questions or for any clarifications. In the next part we will discuss 3-dimensional coordinate systems, and why they are important with respect to measuring and manipulating vector quantities, and how this leads to orthogonal systems of "basis vectors" for any given coordinate system or frame of reference.

RMT
 
creedo,

For this thread can we hold off on the "stringed space" stuff? Please?

Rainman is approaching this, at this point in time, from the POV of basic, vaniilla flavored, no particular complications high school physics. We need to build the stage in such a way that people can understand the subject one plank at a time without jumping into the ozone and leaving the "details" behind.

Thanks. /ttiforum/images/graemlins/smile.gif
 
Rainman,

Great start! Thanks.

Discussion:

At this point in the discussion we can make a generalization (that holds true for now but not necessarily as we move into higher and more complicated areas of the description of physical reality):

A scalar is generally described by one number - the magnitude - and it is a coordinate.

A vector is described by two or more numbers (depending on the number of dimensions) - it is described by a series of coordinates that describe both the direction and magnitude. What makes a vector interesting is that if you rotate the frame of reference the individual coordinates change (in 3D space, x,y,z,) but the vector does not change. The vector itself has, as a whole, a more "meaningful" reality that its individual components.

I won't say any more because coordinate systems is Rainman's next lecture.
 
is a rank 3 tensor the same as a vector with 3 dimensions? I'm familiar with all of the above but the term tensor I never heard a lot of until now.
 
newbie,

is a rank 3 tensor the same as a vector with 3 dimensions? I'm familiar with all of the above but the term tensor I never heard a lot of until now.
Click to expand...

No. The rank of the tensor indicates the number of indexes (indices), i.e. A_ij where "A" has two indexes "i" and "j". But were jumping the gun.
 
I think I used a term incorrectly (dimension of vector).

Reformed question:

<1,2,3>

I'm certain the above is a vector (probably an ambigious term but that's the only way it's been explained to me until now) but is it also considered a rank3 tensor?
 
With the recent "Physics Question" thread, and the subsequent questions that have come up there, the answers to those questions (Lorentz Transformations) are getting back to the topic of vectors and tensors.

Therefore, perhaps it is TIME to move on to the next level of explanation of vectors, and that would be to introduce the concept of COORDINATE SYSTEMS and how they result from our need to define a Vector as being a MAGNITUDE and DIRECTION. We need to be able to accurately describe what we mean by the concept of a vector's DIRECTION. In this post we will define the spatial coordinate systems that are used to define the fields of action that a vector operates within. We will describe and establish the mathematical foundation for coordinate systems by progressing from 1-dimensional, to 2-dimensional, and then finally to 3-dimensional spatial, vector coordinate systems. So let's gets started with the simplest vector, which only operates in 1 spatial dimension/direction...

1-Dimensional Vector Coordinate System (+ or -) "X"
The description of a 1-dimensional vector and its associated coordinate system is quite easy to picture, especially since we always will draw a vector as a line with an arrowhead at one end. The length of the line we draw for a vector represents the vector quantity's Magnitude while the direction that the arrowhead is pointing in represent's the vector quantity's Direction. The easiest way to visualize this is by simply constructing a number line which starts with zero in the middle, and extends towards infinity in both directions (+ and -) away from zero. We can then directly overlay the classical drawing of a vector (the line with the arrowhead) on top of the number line. The Magnitude of the vector then becomes which numerical value on the number line that the head of the vector stretches to touch. The Direction of this 1-D vector is then described as either "+X" or "-X" depending on whether the vector stretches from zero in the positive direction along the number line, or in the negative direction. This simple way to visualize a 1-D vector helps us set the stage for how we will handle vectors in higher dimensional, orthogonal coordinate systems.

2-Dimensional Vector Coordinate System (+ or -) ("X" & "Y")
To move on to a description of a 2-dimensional vector coordinate systems, we simply add another number line, except we must rotate it such that it is 90 degrees (orthogonal) to our original, 1-D "X" axis. Of course, we call this second dimensional axis the "Y" axis. But that is just about where the simplicity ends, as once we move to a 2-D space, we must now have the mathematical tools of trigonometry to help us in dealing the vectors that can be oriented in any direction of this 2-D space. Certainly we can take the vector arrow diagram and still line it up with the "X" axis, but now we can also line the vector arrow diagram up with the "Y" axis as well. And that is not all, since we can draw the vector arrow at any 2-D orientation with respect to the "X" and "Y" axes. For example, if we had a vector quantity (say a force) with a Magnitude of 5 pounds, but a Direction of 45 degrees with respect to the X and Y axes, we can see that not all of the 5 pounds of force is going in the "X" direction, and not all of it is going in the "Y" direction. With the help of trigonometry, which finds its basis in the Pythagorean Theorem, we can calculate that the overall force of 5 pounds acting at a 45 degree angle to the X and Y axes will yield a force of approximately 3.54 pounds in the "X" direction and another 3.54 pounds in the "Y" direction. Now some would say "Hey, 3.54 lbs + 3.54 lbs does NOT add up to 5 pounds!" This is correct, and the reason is because these components operate in orthogonally different directions in this 2-D space, we must use the Pythagorean Theorem to add their effects to get the resulting force. This is called Vector Addition of orthogonal vector components. If you apply the Pythagorean Theorem to the X and Y components of the force (3.54 pounds in each axis), what you end up with is:

Resultant Vector Magnitude = SQRT(X-magnitude^2 + Y-magnitude^2) = SQRT(25) = 5 Pounds!

This is why trigonometry is an essential mathematical skill for a scientist, and especially for an engineer. Since we need to keep track of the "components" of vectors in an orthogonal coordinate system, we must be able to apply the helpful tools of mathematics to break down a vector with a resulting Magnitude and Direction into the component vector Magnitudes in the individual coordinate axes Directions . Doing this makes it easy to add many different vectors with many different directions. We simply break each vector down into its "X" and "Y" components, and then add all the "X" components together, separately add all the "Y" components together, and then we can use our trigonometry skills in a reverse fashion to calculate the overall resulting vector Magnitude and Direction of the summed vector components.

3-Dimensional Vector Coordinate System (+ or -) ("X" & "Y" & "Z")
OK, if we have made the leap of 2 dimensions, and we can handle using the tools of trigonometry reqired to deal with vectors in 2 dimensions, then the extension to a full-up, 3-dimensional coordinate system is as easy as adding a 3rd number line. And this 3rd number line must be at 90 degrees (mutually orthogonal) with respect to the original "X" and "Y" axes. Hence, we have arrived at a complete 3-D "X-Y-Z" orthogonal coordinate system. Now, we can also see that there is going to be a LOT more trigonometry that will need to be performed, because now a vector can take on any orientation in 3-D space with respect to the X,Y, and Y coordinate axes. Just as in the 2-D system, we use the trigonometry to decompose a single vector's Magnitude and Direction into the vector's components in the individual X, Y, and Z directions.

The material we have covered here is some of the most important, early fundamental knowledge that an engineering student must learn, understand, and show proficiency in. This is typically done in a course called "vector statics" which is often taught as part of the mechanical engineering curriculum. This is one of the earliest classes that begins to "wash out" those students who do not have the math skills (or do not wish to acquire them) that are needed to move to more advanced science and engineering topics. The ability to work in coordinate systems with vector statics concepts is a foundation for the next most complicated course which is aptly named "vector dynamics".

As we close this section of this thread, it is important to note that there are many more orthogonal coordinate systems than just the "X-Y-Z" system (commonly called a Cartesian coordinate system). We can also define another popular orthogonal 3-D coordinate system which is especially useful for working problems related to planets, stars, and moons. We call this orthogonal coordinate system a Spherical coordinate system since the coordinate system is centered on spheres rather than straight X-Y-Z "cubes". Instead of "X-Y-Z", in the Spherical coordinate system we use three different geometric variables, as follows:

Rho = Radius (The distance from the center of a sphere to some points on its outer surface).
Theta = Azimuth (The angular displacement from a given reference line around the horizontal equator of the sphere)
Phi = Elevation (The angular displacement from a defined polar axis of the sphere which is orthogonal to the equator's reference horizontal line)

That's enough for now. I think I should pause again, and allow folks to ask questions or perhaps point out errors or omissions of important concepts that I may have committed in this writeup. In the next writeup, I will begin to introduce the concept of vector mathematics, specifically focusing on the two different means to multiply vectors and/or scalars: The "vector dot product" and the "vector cross product".

RMT
 
Hi Newbie,
Reformed question:

<1,2,3>

I'm certain the above is a vector (probably an ambigious term but that's the only way it's been explained to me until now) but is it also considered a rank3 tensor?
Click to expand...
The answer is no. And as Darby alluded to, we are jumping the gun a bit. As we move into tensors it will become clear to you why the answer is no. But until then just remember the rule of thumb that a vector is, by definition, a tensor of rank 1, and a scalar is, by definition, a tensor of rank 0.

The vector example you have given above is still a rank 1 tensor; however, it is a vector that describes a 3-dimensional space. The following is also a vector (rank 1 tensor):

<1,2,3,4,5>

The difference (if you assume that these 5 numbers all represent magnitudes in different, orthogonal dimensions) is that this vector describes a 5-dimensional space.

Got it?
RMT
 
Hi fletch,

Next lesson Mr.Rmt
Click to expand...
OK. I was not aware that there were people still interested. Perhaps this weekend, while I am working on lesson plans for class next week, I can throw together a post to discuss vector multiplication and how the "dot product" differs from the "cross product" and how the concept of a generalized tensor arises from the consideration of these mathematical operations on vectors.

RMT
 
You teach, and you are an aerospace engineer?
Click to expand...
Yes. And I apologize for not getting the next lesson up here. This weekend was especially hectic, as is this week at work. If I am lucky, I'll have more time over Super Bowl Weekend! /ttiforum/images/graemlins/smile.gif

RMT
 
I'm caught up now. Im still not sure how to use a tensor but I understand that a vector can be a represented by a tensor, and tensors can represent much more than a simple vector (right?).

Basically, if I were to make a software program to do tensor math and vector math, I could use the tensor class to make a vector class and re-use a lot of the code. (although it might not be a computationally efficient way to set it up, I'm just trying to understand the differences).

As for vectors, magnitude, direction, dot products etc, I'm familiar with how to use all of that from programming in Second Life and hardware shader languages.
 
So I'm very interested in how to multiply a rank 3 tensor by a 3d vector as in (m * s^3). How would I do that? :-)
Click to expand...
Well, before we can discuss how to do that we need to get to the discussion of covariance and contravariance, and what they mean to vectors, tensors, and coordinate system transformations in general. Whether a particular tensor's indices are covariant or contravariant makes a big difference to the transform you are performing, the physical situation it describes, and what operations are "admissable".

And before we introduce and discuss covariance and contravariance, we should describe and discuss the tensor (matrix) operations for multiplication: Dot Product and Cross Product. That is going to be my next "lecture" in this thread once I can muster the block of time to organize my thoughts and review my notes/math. I apologize for not getting to it sooner, but school has been demanding this quarter and I just gave midterms last week and am grading them all this weekend. I won't be teaching the spring quarter at all, so this will give me time to get a few more lectures in this thread...at least up to explaining co/contra variance. Of course, Wikipedia is a good place to start reading to become familiar with these concepts if you think you are capable enough with Dot and Cross Product operations.

Other mathematical operations and concepts (e.g. the Kroneker Delta, the Metric Tensor, etc.) arise out of these discussions which have special rules and perform useful transformations.

But let me try to start addressing your question: OK, so you understand that the "s^3" part of my equation represents a 3-vector. One example of how to represent this is as (sx, sy, sz). We can also use numerical indices (s1, s2, s3). So a vector has one covariant index... assign it the variable letter "i". This allows us to refer to the vector "s" by the notation "s(i)" or as we use it in the equation "s(i)^3". Where "i" ranges from 1-3.

Now let's deal with Mass/Matter and my "special form of mass" used in my equation. The way I am treating these quantities mathematically in my theoretical work is as follows:

"Mass" in Newton's F=ma is a vector (rank 1 tensor), so let's assign it the symbol small m and its relative covariant indices: m(i), i=1,2,3

"Matter" in Einstin's E=mc^2 is a rank 2 tensor, so let's assign it the symbol capital M. Since it is rank 2 it has two sets of indices, hence: M(i,j), i/j=1,2,3

Now we get to my "special kind of mass" in my equation, which is a rank 3 tensor. Just to distinguish it as something over and above Matter, let's give it the symbol double capital MM. And its indexing format would be: MM(i,j,k), i/j/k=1,2,3

So MM is a 3x3x3 "cube" of 27 cells. And we must then project the cubic volume defined by s(i)^3 onto this cube. Now if we ignore the issue of whether an index is covariant or contravariant, we could represent my equation in a tensor format as:

Information = MM(i,j,k) * s(i)^3

Are you with me so far?
RMT
 
Back
Top