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The Outer Product and Tensor Transformations

Torque and Angular Momentum

The simulation demonstrates that torque changes a spinning object's angular velocity and angular momentum. Applying a positive torque increases the rotation, while a negative torque slows it down or reverses it. This directly illustrates the rotational dynamics equation $\tau=I \frac{d \omega}{d t}$, where $\tau$ is torque and $I$ is the moment of inertia. Animated result

Moment of Inertia

Moment of inertia is an object's resistance to rotational motion, which depends entirely on how its mass is distributed around the axis of rotation. A lower moment of inertia—like when mass is concentrated near the axis—makes an object easier to spin, requiring less energy for a given angular velocity. Conversely, if the mass is spread out, the moment of inertia is higher, making it harder to rotate. Animated result

Invariant Tensors

The Kronecker delta tensor is an invariant tensor, meaning its components do not change even when the coordinate system is transformed through rotation, scaling, or shearing. This immutability is a defining characteristic of a tensor, distinguishing it from simpler mathematical objects like vectors or matrices whose components are dependent on the chosen coordinate system. Animated result

Vectors and Coordinate Systems

The physical world is independent of the coordinate system used to describe it. A vector field is a physical reality, but its mathematical representation (its components) changes depending on the coordinate system chosen. For example, a radially symmetric field is simple and constant in polar coordinates but complex and variable in Cartesian coordinates. This shows that selecting the correct coordinate system can simplify a problem, reflecting the language used to describe the physics rather than the physics itself. Animated result

Operations and Properties of Tensors

1.Angular Momentum and Angular Velocity

Unlike linear motion, where momentum and velocity are always aligned, angular momentum and angular velocity aren't always in the same direction. This is because of the moment of inertia tensor, which accounts for an object's mass distribution. For asymmetrical objects, this tensor transforms the angular velocity vector, causing a misalignment between the two. Animated result

2.Magnetic Field and Lorentz Force

The magnetic field's anti-symmetric tensor representation is why the Lorentz force is always perpendicular to both the magnetic field and the charged particle's velocity. This mathematical structure—specifically, the zero diagonals and opposing off-diagonal pairs—directly explains the force's rotational effect. Animated result

3.The Stress Tensor

The stress tensor is a linear operator that links a surface's orientation to the force acting on it. It shows that the force vector is not fixed but changes in direction and magnitude depending on the surface's orientation. This predictable relationship is a core characteristic of a tensor. Animated result

4.Tensor Operations and Rank

Tensor operations like the outer product and contraction fundamentally change a tensor's rank. An outer product can combine two vectors (rank 1) into a matrix (rank 2), while contraction can reduce that matrix to a scalar (rank 0). This demonstrates that tensors are not just arrays but objects defined by these transformative operations. Animated result

5.Tensor Transformations and Symmetries

When a tensor is visualized as a transformation that warps a sphere, it reveals that a symmetric tensor causes stretching and compression, while an anti-symmetric tensor causes rotation. This is because any linear transformation can be broken down into symmetric (stretch and shear) and anti-symmetric (rotation) components. Animated result

6.The Quotient Law

The quotient law demonstrates how tensors maintain their relationships across different coordinate systems. If the relationship between two vectors, T and S , is defined by a matrix Q ( $T=Q \cdot S$ ), then for this relationship to hold true when the coordinate system is rotated, the matrix Q must transform according to the rules of a tensor. Animated result

The Metric Tensor Covariant Derivatives and Tensor Densities

The Metric Tensor

The metric tensor is a fundamental tool that acts as a blueprint for a coordinate system's geometry. It defines how to measure distances and angles, especially in non-Cartesian systems. Its off-diagonal components become non-zero in skewed, non-orthogonal grids, while in curvilinear systems like polar coordinates, its components can change with position. The inverse metric tensor is vital for calculations, allowing you to convert between different vector forms while ensuring physical consistency. The simple identity matrix in Cartesian coordinates is a special, simplified case of the metric tensor. Animated results: One || Two || Three || Four || Five

Christoffel Symbols

Christoffel symbols measure how a coordinate system's basis vectors change from point to point. In a simple linear system, they are zero because the basis vectors don't change. However, in curvilinear systems like spherical coordinates, the basis vectors change direction and/or magnitude, which is precisely what the non-zero Christoffel symbols account for. These symbols are essential for calculating the covariant derivative, which correctly captures how a vector changes in a curved or non-linear space. Animated results: One || Two || Three

Covariant Derivative

For a scalar field, the covariant derivative is equivalent to the standard directional derivative and the gradient vector. This is because scalars are simple values and don't change with the coordinate system. For vectors, however, the covariant derivative is more complex, as it must account for changes in both the vector's components and the basis vectors themselves. Animated results: One || Two

Tensors vs. Tensor Densities vs Jacobian and Conservation

A true tensor is a physical quantity that remains independent of the coordinate system; its underlying value is constant. A tensor density, in contrast, is tied to the coordinate system. Its value changes with the Jacobian, which measures how the coordinate volume is scaled during a transformation. This relationship, $\bar{\rho}=\rho \cdot J$, shows that a scalar density changes in proportion to the scaling of the coordinate system itself. The Jacobian is crucial for ensuring that physical quantities are conserved when changing variables. It provides the necessary scaling factor to maintain consistency between different mathematical descriptions, ensuring that the total quantity, such as the number of particles, remains the same. Animated results: One || Two || Three || Four || Five

Kronecker Delta

The Kronecker delta, $\delta_j^i$, is a mathematical tool that represents a geometric operation. When its indices are the same, it acts as an identity transformation, preserving space. When they are different, it acts as a collapsing transformation, reducing space to zero. The generalized Kronecker delta extends this idea, serving as a logical filter to determine if two sets of indices are identical, a permutation, or something else entirely. Animated results: One || Two