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Scalars, Vectors, and Tensors

When dealing with physical quantities there are expressions that imbue on the quantities more information, these are called tensors (which includes vectors). The concept of vectors were developed in the 1800s and then tensors were developed in the 1900s, so yes they are a new concept if you consider new more then a century ago. Tensors were very important in the development of engineering and physics. *So what is a scalar, vector, and tensor in simple terms? *

- Scalar - a single quantity with no direction
- Example: My house is 50 km from here
- Physics: 50 km
- Tensor of rank 0

- Vector - a magnitude and direction
- Example: My house is 50 km NNE from here
- Physics symbol example: \(\vec{x}\)
- Tensor of rank 1

- Tensor - a magnitude with a "stress" associated with it
- Example: My house has a moment of inertia of I
_{ij}for i = x,y,z and j=x,y,z (see matrix below) - Physics symbol example: \(\overleftrightarrow{x}\) (note that there are many different expressions for tensor, the authors prefer the double arrow)
- Tensor of rank 2 or above (yes, engineers do use tensors but in general not until junior or senior year; engineers generally only go to rank 2 but scientists go higher)

- Example: My house has a moment of inertia of I

For tensors of rank 1 and higher a matrix (or vector if you wish) can be used to express them. The matrix expression gets more complicated for rank 3 and 4 and will not be discussed herein (but you can research on your own if you like).

For a vector we can express it as

\[\vec{x} = \left(\array{x \\ y \\ z}\right) = \left(\array{x_1 \\x_2 \\ x_3}\right) = \left(\array{i \\ j \\ k}\right)\]

For a tensor of rank 2 (moment of inertia) we can express it as

\[\overleftrightarrow{I}= \left(\array{I_{xx} & I_{xy} & I_{xz} \\ I_{yx} & I_{yy} & I_{yz} \\ I_{zx} & I_{zy} & I_{zz}}\right)\]

## The math of tensors (scalars,vectors, and tensors)

The mathematics of scalars is obvious and we will not discuss that here, but the math of vectors is less obvious. There are graphical ways to do the addition of vectors, but here we will only discuss the arithmetic way (we will do some graphical methods in the coming sections).

Let us define the a vector A that will consist of three components in Cartesian coordinate system (x,y,z). When defining vectors we define unit vectors as one unit in magnitude of that particular vector (so the equivalent of 1 in scalar form). Here we will define the unit vectors in the x,y,z respectively as \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\). As an example, a vector going in the x-direction of magnitude 5 km would be represented as \(\vec{A} = 5 \hat{i}\). To add to the vector A we will define the vector B as well.

\[\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}\]

\[\vec{B} = B_x\hat{i} + B_y\hat{j} + B_z\hat{k}\]

unit vectors are added and associated with the unit vector:

\[\vec{C} = \left(A_x+B_x\right)\hat{i} +\left( A_y+B_y\right)\hat{j} + \left(A_z+B_z\right)\hat{k} = C_x\hat{i}+C_y\hat{j}+C_z\hat{k} \]

The more interesting operations come with what could be viewed as multiplication of vectors. This is because there are at least three ways to "multiply" the vectors: the dot product, the cross product, and the dyadic vector product.

The **dot product** is the product of two vectors and produces a scalar. From a component view the main rules are that the dot product of same unit vectors are equal to one and different unit vectors are zero. So \(\hat{i}\cdot\hat{i} = 1\) and the like and \(\hat{i}\cdot\hat{j} = 0\). The dot product the vector A and vector B:

\[\vec{A}\cdot\vec{B} = A_x B_x + A_y B_y + A_z B_z\]

The dot product is known as a scalar product and is invariant (independent of coordinate system).

An example of a dot product in physics is mechanical work which is the dot product of force and distance:

\[W = \vec{F}\cdot\vec{d}\]

The **cross product** is the product of two vectors and produce a vector. From a component view the main rules are that the cross product of the same unit vectors are equal to zero and different unit vectors are the cyclic. So \(\hat{i}\times\hat{i} = 0\) and the like and \(\hat{i}\times\hat{j} = \hat{k}\), \(\hat{j}\times\hat{i} = -\hat{k}\) and the like.

\[\vec{A}\times\vec{B}= (A_yB_z - A_yB_z)\hat{i} + (A_xB_z - A_zB_x)\hat{j} + (A_xB_y - A_yB_x)\hat{k}\]

or using Levi-Civita symbols and using \(\vec{x_i}\), \(A_i\), \(B_i\) formalism instead of \(\vec{i}\), \(\vec{j}\), \(\vec{k}\) formalism.

\[\vec{A}\times\vec{B} = \epsilon_{ijk} \vec{x_i} A_j B_k\]

where \( \epsilon_{ijk}\) is **zero** if 2 or more indices are the same and **one** if the indices are a cyclic (clockwise depending on reference) permutation and **negative one** if they are an anticyclic (counter-clockwise) permutation.

The cross product can be though of as a measurement of "co-alignment" of vectors. Some examples:

- Cross product is a directional area (this is very useful): \(d\vec{A} = \hat{n}dA = d\vec{x}\times d\vec{y}\) where \(\hat{n}\) is the unit normal vector (to the Area)
- Cross product is moment of force (torque): \(\vec{\tau} = \vec{r}\times\vec{F}\)

Directional vector x and y can when taken as a cross product results in a directional area which is normal to the directional vectors. The normal vector is shown for reference. | A lever arm (wrench) has a force on it that rotates the screw into the page (or piece of wood). |

The **dyadic cross product** is the product of two vectors and produce a tensor (rank 2). The best way to look at this is through matrices.

\[\overleftrightarrow{C} = \vec{A}\otimes\vec{B} = \left(\array{A_x \\A_y \\ A_z}\right) \left(\array{B_x & B_y & B_z}\right) \hat{\hat{c}} = \left(\array{A_{x}B{x} & A_{x}B_{y} & A_{x}B_{z} \\ A_{y}B_{x} & A_{y}B_{y} & A_{y}B_{z} \\ A_{z}B_{x} & A_{z}B_{y} & A_{z}B_{z}}\right) \hat{\hat{c}}\]

An example of a tensor product would be found in the Navier-Stokes equations which is used in fluid motion (air flow around wings, etc. would be an engineering application). This concept is beyond the scope of these work, so we will stop here on the idea of multiplication of vectors.

*If you can multiple can you divide, right?* Well technically yes but in this case the definable quantities are not unique. If there is some physics that involves division in this realm it probably is better going to the slightly older quaternion system (the vector system offered some improvements to quaternions, but quaternions are still used today especially with regards to rotation).

## Fields

The idea of fields can lead to confusion when first learning the idea of vectors. Fields are not vectors or tensors but may contain them or be derivable from them.

- A scalar field is a field of scalar points
- Measurements of the amount of rain in a field at certain points would be an example of a field of scalar points
- This is NOT a vector

- A vector field is a field of vectors
- This is not a tensor

- A tensor filed is a field of tensors

## Important differentials with vectors

We add this for completeness without going into detail so as to introduce a concept for latter use.

There are differential transforms that involve vectors as well. This operations (or transforms) are very important in describing physical process that best described with vectors. The are applicable to engineers, scientists, economists, etc.

There is the gradient of a "scalar" function which produces a "vector" function. The gradient is exactly like it is in just regular English (going up a steep hill has a large gradient and going up a slow rising hill has a small gradient). In this context it is a vector measurement of the change of a "scalar" function. Given a function f(x,y,z) the gradient is \(\vec{\nabla} f\). The \(\nabla\) represents the component partial differential of the function which you will learn later in your academic career. The vector symbol is used to indicate that each component will be associate with a unit vector. Examples: **force** is the gradient of potential energy and the **electric field** is a gradient of the electric potential field.

The divergence of a vector function produces a scalar function. Here again regular English applies as this operation (transform) gives a result that describes divergence of a vector function. This is very useful in measuring the characteristics of a sink or source (or if you have a sink or source at all). Given a vector function the divergence is \(\vec{\nabla}\cdot\vec{F}\). The most famous example of this are the divergence of the electric field being related to the charge density and the divergence of the magnetic field being equal to zero. *How are these famous?* They are two of Maxwell's equations (in differential form). Which simplistically says there are charges that produce an electric field, but no magnetic "charges."

The curl of a vector function produces a vector function. Here again regular English applies as this operation (transform) gives a result that describes the curl (or circular density) of a vector function. This gives an idea of rotational nature of different fields. Given a vector function the curl is \(\vec{\nabla}\times\vec{F}\). The most famous example of this is the curl of the electric field being related to the magnetic field and visa versa. *How are these famous?* They are the other two of Maxwell's four equations.

Note in the divergence the dot product is used and in the curl the cross product is used as defined previously. The differential extensions are clearly very important and will be fully introduced in later courses.

Lastly there is a more complicated operation (transform) called the Laplacian that has various forms: one for scalars, one for vectors, and one for tensors of higher rank. This uses the nabla^{1} \(\nabla\) and dot product and cross product to produce a measure of distribution or diffusion. Not all engineers will need to be familiar with this so we will stop here.

^{1}Nabla is Ancient Greek for Phoenician harp as it somewhat represents a harp. This operator is called the Del operator.