Vectors and Tensors - Demosophy.org

Vectors

In the three-dimensional Cartesian coordinate system with a Euclidean metric, the gradient, if it exists, is given by:

\nabla f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} ,

where $i$ , $j$ , $k$ are the standard unit vectors in the directions of the $x$ , $y$ and $z$ coordinates, respectively. For example, the gradient of the function

f(x,y,z)=2x+3y^{2}-\sin(z)

\nabla f=2\mathbf {i} +6y\mathbf {j} -\cos(z)\mathbf {k} .

If the function

f

is differentiable, the gradient of

f

at a point is either zero, or perpendicular to the level set of

f

at that point.

To understand what this means, imagine that two hikers are at the same location on a mountain. One of them is bold, and he decides to go in the direction where the slope is steepest. The other one is more cautious; he does not want to either climb or descend, choosing a path which will keep him at the same height. In our analogy, the above theorem says that the two hikers will depart in directions perpendicular to each other.

$If$

$z=f(x,y)$

$F(x,y,z)=z-f(x,y)=0,$

$\mathbf {n} =\nabla F(x,y,z)=\left(-{\tfrac {\partial f}{\partial x}},-{\tfrac {\partial f}{\partial y}},1\right).$

Inner multiplication or Dot product

https://en.wikipedia.org/wiki/Exterior_algebra#Inner_product

Cross product or Vector product

The cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space is denoted by the symbol X.
Given two linearly independent vectors $a$ and $b$ , the cross product, $a \times b$ (read “a cross b”), is a vector that is perpendicular to both $a$ and $b$ ,^[1] and thus normal to the plane containing them.

https://en.wikipedia.org/wiki/Cross_product

The exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v denoted by
$u\wedge v,$ is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original
space of vectors. The magnitude^[4] of $\dpi{100} \large {\displaystyle u\wedge v,}$ can be interpreted as the area of the parallelogram with sides u and v which in three dimensions can also be computed using the cross product of the two vectors.

An eigenvector (/ˈaɪɡənˌvɛktər/) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by $\lambda$ , is the factor by which the eigenvector is scaled.

In rotational motion of a rigid body, the principal axes are the eigenvectors of the inertia matrix.

real symmetric matrices have real eigenvalues.

If A is an n × n matrix, then the sum of the n eigenvalues of A is the trace of A and the product of the n eigenvalues is the determinant of A.

Also

Suppose a plane contains two vectors $u$ and X $v .$ The normal to the plane, the unit vector is determined by the formula

$\dpi{120} \large \boldsymbol {\vec{n}=\frac{\vec{x}_u\times\vec{x}_v}{|\vec{x}_u\times\vec{x}_v|}}$

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Cross x product In Physics

angular velocity $ω$ was defined as the time rate of change of angle $θ :$

$ω =$ $\boldsymbol{\frac{\Delta\theta}{\Delta{t}}},$

where

θ

is the angle of rotation as seen in Figure 1. The relationship between angular velocity

ω

and linear velocity

v

was also defined in Chapter 6.1 Rotation Angle and Angular Velocity as

v = r ω

ω =

\frac{v}{r},

where $r$ is the radius of curvature, also seen in Figure 1.

The instantaneous angular velocity ω at any point in time is given by

$\boldsymbol {\vec{\omega} =\frac { ( \vec{r} \times \vec{v} ) } {r^{2}} = {\frac {\vec{v_{\perp }}}{r}}}$

where $r$ is the distance from the origin and $v_{\perp }$ is the component of the instantaneous velocity that is perpendicular to the position vector $\huge ^{\vec{r}}$ ,

$v_{\perp }$ by convention is positive for counter-clockwise motion and negative for clockwise motion.

ω is a pseudovector that resembles a vector, and behaves like a vector in many situations, but is changed into its opposite if the orientation of the space is changed.

angular acceleration refers to the time rate of change of angular velocity. As there are two types of angular velocity, namely spin angular velocity and orbital angular velocity, there are naturally also two types of angular acceleration, called spin angular acceleration and orbital angular acceleration respectively. Spin angular acceleration refers to the angular acceleration of a rigid body about its centre of rotation, and orbital angular acceleration refers to the angular acceleration of a point particle about a fixed origin.

Angular acceleration is measured in units of angle per unit time squared (which in SI units is radians per second squared), and is usually represented by the symbol alpha (α).

Therefore, the instantaneous angular acceleration α of the particle is given by^[2]

\alpha ={\frac {d}{dt}}\left({\frac {v_{\perp }}{r}}\right).

https://en.wikipedia.org/wiki/Angular_acceleration

Example1: Torque or the First Moment of Force

The net ”torque” on a point particle is defined to be the pseudovector

$\boldsymbol{\tau} = \mathbf r \times \mathbf F$

$\vec{\tau} = \vec{r} \times \vec{F}$

$\LARGE \boldsymbol {= (r_y F_z - r_z F_y) \vec{i} + (r_x F_z - r_z F_x) \vec{j}+ (r_x F_y - r_y F_x) \vec{k}}$

$\tau =\|\mathbf {r} \|\,\|\mathbf {F} \|\sin \theta \,\!$

$\tau =\|\mathbf {r} \|\,\|\mathbf {F_{\perp}} \|\$

where

${\boldsymbol {\tau }}$ is the torque vector and $\tau$ is the magnitude of the torque,
$\mathbf {r}$ is the position vector (a vector from the point about which the torque is being measured to the point where the force is applied),
$\mathbf {F}$ is the force vector,
$\times$ denotes the cross product, which produces a vector that is perpendicular to both $r$ and $F$ following the right-hand rule,
$\theta$ is the angle between the force vector and the lever arm vector.

In the special case of constant distance $r$ of the particle from the origin ${\tfrac {dr}{dt}}=0$

{\boldsymbol {\tau }}=mr^{2}{\boldsymbol {\alpha }},

which can be interpreted as a “rotational analogue” to $\mathbf {F} =m\mathbf {a}$ , where the quantity $mr^{2}$ (known as the moment of inertia of the particle) plays the role of the mass $m$ . However, unlike $\mathbf {F} =m\mathbf {a}$ , this equation does not apply to an arbitrary trajectory, only to a trajectory contained within a spherical shell about the origin.

https://en.wikipedia.org/wiki/Torque

Example2: Angular Momentum or the First Moment of Linear Momentum

angular momentum is the 1st moment of momentum: $\mathbf{L} = \mathbf{r} \times \mathbf{p}$ . Note that momentum itself is not a moment. it is a pseudovector.

It is the cross product of the particle’s position vector $r$ (relative to some origin) and its momentum vector; the latter is $p = m v$ in Newtonian mechanics. Unlike linear momentum, angular momentum depends on where this origin is chosen, since the particle’s position is measured from it.

Similar to conservation of linear momentum, where it is conserved if there is no external force, angular momentum is conserved if there is no external torque.

Angular Momentum can be calculated for non-circular motion if only the component of the motion which is perpendicular to the radius vector is considered. In that case,

L=rmv_{\perp },

where $v_{\perp }=v\sin(\theta )$ is the perpendicular component of the velocity to r.

${L} =\mathbf {r} \times \mathbf {p}$

${L} =\mathbf {r} \times \mathbf {mv}$

$=m ( {r} \times {v} )$

${L} =\left(r^{2}m\right)\left({\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}\right)\\$

Which is Shown as as:

\mathbf {L} =I{\boldsymbol {\omega }},

where

$I=r^{2}m$ is the moment of inertia for a point mass,
${\boldsymbol {\omega }}={\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}$ is the orbital angular velocity of the particle about the origin,
$\mathbf {r}$ is the position vector of the particle relative to the origin, and $r=\left\vert \mathbf {r} \right\vert$ ,
$\mathbf {v}$ is the linear velocity of the particle relative to the origin, and
$m$ is the mass of the particle.

Ref1 Book

Tensors:

A tensor is an n-dimensional array satisfying a particular transformation law.

Not every Matrix is a rank 2 Tensor.

A Rank 2 tensor can be represented as a matrix of numbers — in conjunction with an associated transformation law.

A Rank 3 tensor can be represented as a 3-dimensional array of numbers — in conjunction with an associated transformation law.

https://rukshanpramoditha.medium.com/real-world-examples-of-0d-1d-2d-3d-4d-and-5d-tensors-100b0837ced4

Rank 3 example: Levi-Civita_symbol

https://en.wikipedia.org/wiki/Levi-Civita_symbol

Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).

A covariant tensor, denoted with a lowered index (e.g., a_mu) is a tensor having specific transformation properties.

A contravariant tensor is a tensor having specific transformation properties (cf., a covariant tensor).

https://rinterested.github.io/statistics/tensors2.html

https://www.mathpages.com/rr/s5-02/5-02.htm

https://math.stackexchange.com/questions/8170/intuitive-way-to-understand-covariance-and-contravariance-in-tensor-algebra