# Operations and Properties of Tensors

Tensor algebra, which includes vector-like operations such as addition and scalar multiplication, defines key products like the outer product, which creates a higher-rank tensor, and the contracted product, a generalization of the inner product formed by contraction, an operation that reduces rank by two. A tensor's inherent properties, like symmetry or anti-symmetry, allow for a unique decomposition of any rank-two tensor, while the Quotient Law is a crucial theorem used to verify if a mathematical object behaves as a true tensor.

### :clapper:calculate and display the angular velocity vector and the resulting angular momentum vector

> the angular momentum and angular velocity vectors are not always aligned. This is different from linear motion, where linear momentum and velocity always point in the same direction. The reason for this misalignment is the moment of inertia tensor, which accounts for how an object's mass is distributed. When an object is not symmetrical, like our cube with different inertia values on each axis, its resistance to rotation varies depending on the axis. The tensor essentially transforms the angular velocity vector to produce the angular momentum vector, causing them to point in different directions.

{% embed url="https://youtu.be/fCu8LTl3ebM" %}

### :clapper:the force being perpendicular to both the velocity and the field as result of the tensor's anti-symmetric nature

> the anti-symmetric nature of the magnetic field's tensor representation is the fundamental mathematical reason why the resulting force on a charged particle is always perpendicular to both the field and the particle's velocity. It shows how the zero diagonal elements and the opposite-signed off-diagonal pairs in the tensor directly correspond to the rotational effect of the Lorentz force.

{% embed url="https://youtu.be/cllEf9J7Qio" %}

### :clapper:the stress tensor acts as a linear map that transforms the surface normal vector into the force vector

> The stress tensor is a powerful mathematical tool that acts as a linear operator, linking a surface's orientation to the force acting on it. The demo visually demonstrates that the force vector is not a fixed quantity; its direction and magnitude are entirely dependent on the orientation of the surface, as defined by the stress tensor. This consistent, predictable relationship is precisely what defines a tensor.

{% embed url="https://youtu.be/XJMbaPT0EtM" %}

### :clapper:Visualize the outer product and contraction operations on tensors

> The demo effectively visualizes that tensor operations, like the outer product and contraction, fundamentally change a tensor's rank, revealing how two 1D vectors can combine to form a 2D matrix, and how that matrix can then be reduced to a single scalar, illustrating that tensors are not merely arrays of numbers but objects whose properties are defined by these transformative operations.

{% embed url="https://youtu.be/s_mGd2oNdo0" %}

### :clapper:how symmetric and anti-symmetric tensors behave by visualizing their effect on a sphere

> The visualization uses a sphere as a reference shape. The tensor's components define a transformation matrix that warps the sphere into a new shape. You'll see that symmetric tensors create stretches and compressions along specific axes, while anti-symmetric tensors cause a rotation. This is because a general linear transformation can be decomposed into a rotation, a shear, and a stretch—the symmetric part handles the stretch and shear, and the anti-symmetric part handles the rotation.

{% embed url="https://youtu.be/roTavryGC7I" %}

### :clapper:The quotient law of tensors provides a test for whether a given set of components forms a tensor

> The visualization demonstrates that if we rotate our coordinate system, the vectors $S$ and $T$ transform according to the rules of a vector. For the relationship $$T=Q \cdot S$$ to remain true in the new coordinate system, the matrix Q must also transform according to the rules of a tensor. This is the essence of the quotient law.

{% embed url="https://youtu.be/FTSkZGGBhHY" %}

### :bubbles:Cue Column

{% embed url="https://viadean.notion.site/Operations-and-Properties-of-Tensors-25f1ae7b9a328016bc45cacb50256651?source=copy_link" %}
