🎬Animated Results
🎬how the metric tensor changes with the geometry of a coordinate system🎬a non-orthogonal coordinate system dynamically calculating and displaying the metric tensor and its🎬how the metric tensor in polar coordinates is used to compute the circumference of a circle🎬compares a simple linear coordinate system with zero Christoffel symbols to a curvilinear system wit🎬how an orthogonal coordinate system can still have non-zero Christoffel symbols if its basis vectors🎬Directional Derivative equals Covariant Derivative for any scalar field🎬illustrate the relationship between the covariant derivative and the gradient of a scalar field on a🎬how the metric tensor allows for raising and lowering indices by seeing two distinct calculations🎬the impact of the Jacobian on the distributions of both energy and momentum🎬Explain why the permutation symbol isn't a true tensor but is instead a tensor density with a weight🎬The metric determinant is a scalar density with a weight of two and its square root is a scalar dens🎬how a completely anti-symmetric tensor is constructed from a tensor density🎬Visualize the geometric transformation of the Kronecker delta on a cube🎬how the partial and covariant derivatives behave in a polar coordinate system🎬calculate and display the angular velocity vector and the resulting angular momentum vector🎬the force being perpendicular to both the velocity and the field as result of the tensor's anti-symm🎬the stress tensor acts as a linear map that transforms the surface normal vector into the force vect🎬Visualize the outer product and contraction operations on tensors🎬how symmetric and anti-symmetric tensors behave by visualizing their effect on a sphere🎬The quotient law of tensors provides a test for whether a given set of components forms a tensor🎬A disc of mass M and radius R rotating around its symmetry axis with angular velocity🎬The moment of inertia by comparing two discs rotating around different axes🎬the invariance of the Kronecker delta tensor under various coordinate transformations🎬Visualize the curvilinear coordinate through Cartesian grid and a polar coordinate system in a Eucli🎬how a quantity's value changes with a change in the coordinate system by visualizing the difference🎬calculate the value of the generalized Kronecker delta to observe permutation Check and permutation🎬focus on the tangent vector basis and Christoffel symbols in polar coordinates🎬how the Kronecker delta and Christoffel symbols behave in a Cartesian coordinate system🎬visualize the relationship between the angular velocity vector and the angular momentum vector for a🎬focus on how a force defined by the stress tensor acts on a surface resulting in a total force vecto🎬calculates the two non-zero components of the moment of inertia tensor based on the cylinder's prope🎬how changing the external magnetic field affects the forces on a current-carrying wire🎬visualize the buoyant force on a submerged object and how the total force changes by adjusting the d🎬the buoyant force on an object immersed in a fluid with the pressure field is equal to the force on🎬how the stress changes from the top to the bottom of the rod🎬how the force between two charges can be re-imagined as a force mediated by the electric field🎬the high conductivity within the carbon sheets by showing a constant flow of bright and fast-moving🎬compare the free precession of a disc and a prolate spheroid🎬apply different transformations to a grid of particles representing a solid material and observe how🎬visualize elasticity and Hooke's Law for an isotropic material🎬Visualize how vector components change with the coordinate system🎬how the difference in how an electric field propagates through a vacuum versus how it propagates thr🎬The polar coordinate tangent vectors depend on Cartesian basis vectors🎬the polarisation is proportional to the electric field within an isotropic medium that is contrary🎬how the conductivity tensor affects the relationship between the electric field and current density🎬how tensors behave under a change of basis🎬how a tensor's components change when you transform the coordinate system while the result of the co🎬how the components of the tensor and two vectors change as the coordinate system rotates while the f🎬a type (2,0) tensor that is initially symmetric and the transformed tensor remains symmetric🎬how the components of the tensor and two vectors change as the coordinate system rotates while the f🎬Show symmetric and anti-symmetric components of 2D tensor represented by a stretching ellipsoid and🎬A tensor contraction only depends on its symmetric part if the contracted indices are symmetric in🎬visualize a generic type (0, 3) tensor and then show how a zero tensor appears🎬visualize a general type (3,0) tensor as a 3D grid of spheres where each sphere corresponds to a com🎬a tensor as a physical object that exists independently of the coordinate system used to describe it🎬how the magnetic field vector points in the azimuthal direction and how its magnitude changes with d🎬the gradient of a scalar field is a covariant vector🎬the fictitious forces that arise in non-inertial frames of reference🎬how to calculate divergence in a curved coordinate system🎬how the divergence field emerges from the antisymmetric tensor🎬the geometrical meaning of the Christoffel symbols in flat Euclidean space using the non-Cartesian s🎬Cylindrical has one scaled tangent direction while Spherical has two🎬Finding the Arc Length in Spherical Coordinates🎬visualize the underlying geometry and the tangent basis vectors defined by the metric demonstrating🎬display the instantaneous line element at a moving point and compare the standard Cartesian length c🎬a 2D type (1,1) tensor under rotation under both linear transformation and non-linear transformation🎬the covariant derivative is indispensable in non-Cartesian or curved systems that distinguishes it f🎬why the standard divergence formula requires spherical volume element🎬the Laplace-Beltrami Operator measures the local curvature of the field by detecting peaks and troug🎬the divergence of the tangent basis vectors illustrates why these coordinate systems have non-Cartes🎬the square root of the determinant of the metric tensor unifies the Divergence of the Gradient in Cu🎬the physical density is invariant as a geometric object but its coordinate representation changes🎬Jacobian determinant for a composite coordinate transformation is the product of the individual Jaco🎬the metric determinant scales vector operations🎬the totally antisymmetric tensor in a flat two-dimensional space🎬how two dynamic inputs determine the covariant components of the resulting vector🎬Visualizing the Curl of Dual Bases in Curvilinear Coordinates🎬What're geometric actions associated with antisymmetric tensors and symmetric tensors🎬rigid-body motion using an orthogonal affine transformation🎬visualize the density fields of Kinetic Energy Momentum and Angular Momentum as a function of time🎬Visualize the area element (the Jacobian determinant) helps illustrate how the transformation stretc🎬Visualize how the Poisson's ratio approaches the incompressibility limit as the stiffness ratio incr🎬how the magnetic stress tensor decomposes to show that magnetic fields simultaneously exert tension🎬Visualize the electric field lines and the resulting surface forces between the attractive and repul🎬the local Lorentz force density is zero if there are no charges or currents present to act upon🎬stability and complexity of motion are governed by the relationship between the angular velocity and🎬Centrifugal Force as the Stability Governor🎬The centrifugal force depends only on the particle's perpendicular distance in a rotating reference🎬Clarifying the Contributions of the Tidal Tensor Components🎬Visualizing the Magnetic Stress Tensor is highly illustrative of how magnetic fields exert forces🎬Visualize Electric Field Divergence from Charge and Magnetic Field Curl from Changing Electric Field🎬Visualize the behavior of both intensive properties and extensive properties🎬Visualize the conservation of momentum and the action-reaction forces through the two-body collision🎬show the exponential decay over the time scale set by the characteristic charge relaxation time🎬Only the component of the current density vector that is exactly perpendicular to the surface contri🎬the difference between concentration change due to external flow and concentration change due to int🎬water exits a faucet and accelerates under gravity while its velocity increases as it falls🎬how the shape and peak height change for pure diffusion and pure decay and the combined scenario🎬Point Source Diffusion-From Transient Pulse to Steady Source🎬Electrostatic Potentials-Numerical Validation of Point vs Distributed Charge🎬How the delta function is used to model charge distributions concentrated on a line or a surface ins🎬the concentration profile over time for three scenario-pure diffusion and pure convection and the co🎬Fick's second law is used to Chemical Mixing and Heat Transfer and Semiconductor Doping🎬Dissecting the Deterministic Roles of Diffusivity Decay and Dimensionality in PDEs🎬Core Scientific Laws and Thermodynamic Properties Illustrated Through Dynamic Visualization🎬visualize how an initial wave profile splits into two equal and opposite-traveling components🎬visually compares the behavior of an undamped wave and a damped wave over time🎬visualize the wave equation solution for the condition where the string starts with zero initial dis🎬the net forces and tension acting on a small element of a vibrating membrane🎬Dynamic Visualization of Wave Equation Principles-Analyzing Force Balance and Traveling Wave Compone🎬Boundary conditions define the allowed solutions (eigenmodes) and the natural frequencies🎬The demonstration of the nature of transient heat diffusion and the importance of the Fourier number🎬how the Neumann boundary condition dictates the behavior of a sealed system🎬using the Finite Difference Method to solve the 1D wave equation with the mass-loaded boundary condi🎬Steady-State Heat Transfer-Comparison of Dirichlet and Robin (Newton's Cooling) and Neumann Boundary🎬how the choice of boundary condition fundamentally dictates the long-term equilibrium and the result🎬The initial condition is the crucial starting point for any time-dependent simulation🎬Visualize how the string vibrates over time as a superposition of standing waves🎬Visualize solutions to Poisson's Equation and Laplace's Equation🎬how the string's equilibrium is fundamentally shifted by the constant external force🎬the stationary solution to Poisson's equation driven by a Dirac delta function source🎬how the string's equilibrium is fundamentally shifted by the constant external force🎬the stationary solution to Poisson's equation driven by a Dirac delta function source🎬the analytical solution for a specific mode of the Helmholtz equation using Bessel functions🎬Plot the region where the Principal Symbol is zero for both the Wave Operator-Hyperbolic and the Dif🎬Stability Analysis of Reaction-Diffusion Equations-Linearization Demonstrating Growth and Decay🎬the convective transport of momentum through a surface element by a fluid moving with velocity and 5🎬From Dust to D-Force-Visualizing the Cauchy Momentum Equation🎬Visualize hydrostatic equilibrium is best done by focusing on the balance of forces acting on an inf🎬visualize the inverse relationship between fluid speed and pressure along a streamline-Bernoulli's p🎬how pressure and temperature and density and velocity change as a gas flows isentropically through a🎬Hagen-Poiseuille flow-or called Poiseuille flow through a circular pipe🎬the relationship between numerical modeling and analytical solutions-Poiseuille's Law in fluid mecha🎬how the three components—the quasi-static response and the transient response and the steady-state f🎬superposition principle in both electrostatics and wave propagation🎬how the three components—the quasi-static response and the transient response and the steady-state f🎬Heat Conduction with Inhomogeneous Condition and Homogeneous Condition
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