# Animated Results

- [how the metric tensor changes with the geometry of a coordinate system](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-metric-tensor-changes-with-the-geometry-of-a-coordinate-system.md)
- [a non-orthogonal coordinate system dynamically calculating and displaying the metric tensor and its](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/a-non-orthogonal-coordinate-system-dynamically-calculating-and-displaying-the-metric-tensor-and-its.md)
- [how the metric tensor in polar coordinates is used to compute the circumference of a circle](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-metric-tensor-in-polar-coordinates-is-used-to-compute-the-circumference-of-a-circle.md)
- [compares a simple linear coordinate system with zero Christoffel symbols to a curvilinear system wit](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/compares-a-simple-linear-coordinate-system-with-zero-christoffel-symbols-to-a-curvilinear-system-wit.md)
- [how an orthogonal coordinate system can still have non-zero Christoffel symbols if its basis vectors](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-an-orthogonal-coordinate-system-can-still-have-non-zero-christoffel-symbols-if-its-basis-vectors.md)
- [Directional Derivative equals Covariant Derivative for any scalar field](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/directional-derivative-equals-covariant-derivative-for-any-scalar-field.md)
- [illustrate the relationship between the covariant derivative and the gradient of a scalar field on a](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/illustrate-the-relationship-between-the-covariant-derivative-and-the-gradient-of-a-scalar-field-on-a.md)
- [how the metric tensor allows for raising and lowering indices by seeing two distinct calculations](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-metric-tensor-allows-for-raising-and-lowering-indices-by-seeing-two-distinct-calculations.md)
- [the impact of the Jacobian on the distributions of both energy and momentum](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-impact-of-the-jacobian-on-the-distributions-of-both-energy-and-momentum.md)
- [Explain why the permutation symbol isn't a true tensor but is instead a tensor density with a weight](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/explain-why-the-permutation-symbol-isnt-a-true-tensor-but-is-instead-a-tensor-density-with-a-weight.md)
- [The metric determinant is a scalar density with a weight of two and its square root is a scalar dens](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-metric-determinant-is-a-scalar-density-with-a-weight-of-two-and-its-square-root-is-a-scalar-dens.md)
- [how a completely anti-symmetric tensor is constructed from a tensor density](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-a-completely-anti-symmetric-tensor-is-constructed-from-a-tensor-density.md)
- [Visualize the geometric transformation of the Kronecker delta on a cube](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-the-geometric-transformation-of-the-kronecker-delta-on-a-cube.md)
- [how the partial and covariant derivatives behave in a polar coordinate system](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-partial-and-covariant-derivatives-behave-in-a-polar-coordinate-system.md)
- [calculate and display the angular velocity vector and the resulting angular momentum vector](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/calculate-and-display-the-angular-velocity-vector-and-the-resulting-angular-momentum-vector.md)
- [the force being perpendicular to both the velocity and the field as result of the tensor's anti-symm](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-force-being-perpendicular-to-both-the-velocity-and-the-field-as-result-of-the-tensors-anti-symm.md)
- [the stress tensor acts as a linear map that transforms the surface normal vector into the force vect](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-stress-tensor-acts-as-a-linear-map-that-transforms-the-surface-normal-vector-into-the-force-vect.md)
- [Visualize the outer product and contraction operations on tensors](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-the-outer-product-and-contraction-operations-on-tensors.md)
- [how symmetric and anti-symmetric tensors behave by visualizing their effect on a sphere](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-symmetric-and-anti-symmetric-tensors-behave-by-visualizing-their-effect-on-a-sphere.md)
- [The quotient law of tensors provides a test for whether a given set of components forms a tensor](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-quotient-law-of-tensors-provides-a-test-for-whether-a-given-set-of-components-forms-a-tensor.md)
- [A disc of mass M and radius R rotating around its symmetry axis with angular velocity](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/a-disc-of-mass-m-and-radius-r-rotating-around-its-symmetry-axis-with-angular-velocity.md)
- [The moment of inertia by comparing two discs rotating around different axes](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-moment-of-inertia-by-comparing-two-discs-rotating-around-different-axes.md)
- [the invariance of the Kronecker delta tensor under various coordinate transformations](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-invariance-of-the-kronecker-delta-tensor-under-various-coordinate-transformations.md)
- [Visualize the curvilinear coordinate through Cartesian grid and a polar coordinate system in a Eucli](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-the-curvilinear-coordinate-through-cartesian-grid-and-a-polar-coordinate-system-in-a-eucli.md)
- [how a quantity's value changes with a change in the coordinate system by visualizing the difference](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-a-quantitys-value-changes-with-a-change-in-the-coordinate-system-by-visualizing-the-difference.md)
- [calculate the value of the generalized Kronecker delta to observe permutation Check and permutation](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/calculate-the-value-of-the-generalized-kronecker-delta-to-observe-permutation-check-and-permutation.md)
- [focus on the tangent vector basis and Christoffel symbols in polar coordinates](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/focus-on-the-tangent-vector-basis-and-christoffel-symbols-in-polar-coordinates.md)
- [how the Kronecker delta and Christoffel symbols behave in a Cartesian coordinate system](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-kronecker-delta-and-christoffel-symbols-behave-in-a-cartesian-coordinate-system.md)
- [visualize the relationship between the angular velocity vector and the angular momentum vector for a](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-the-relationship-between-the-angular-velocity-vector-and-the-angular-momentum-vector-for-a.md)
- [focus on how a force defined by the stress tensor acts on a surface resulting in a total force vecto](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/focus-on-how-a-force-defined-by-the-stress-tensor-acts-on-a-surface-resulting-in-a-total-force-vecto.md): focus on how a force defined by the stress tensor acts on a surface resulting in a total force vector
- [calculates the two non-zero components of the moment of inertia tensor based on the cylinder's prope](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/calculates-the-two-non-zero-components-of-the-moment-of-inertia-tensor-based-on-the-cylinders-prope.md): calculates the two non-zero components of the moment of inertia tensor based on the cylinder's properties
- [how changing the external magnetic field affects the forces on a current-carrying wire](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-changing-the-external-magnetic-field-affects-the-forces-on-a-current-carrying-wire.md)
- [visualize the buoyant force on a submerged object and how the total force changes by adjusting the d](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-the-buoyant-force-on-a-submerged-object-and-how-the-total-force-changes-by-adjusting-the-d.md): visualize the buoyant force on a submerged object and how the total force changes by adjusting the densities
- [the buoyant force on an object immersed in a fluid with the pressure field is equal to the force on](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-buoyant-force-on-an-object-immersed-in-a-fluid-with-the-pressure-field-is-equal-to-the-force-on.md)
- [how the stress changes from the top to the bottom of the rod](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-stress-changes-from-the-top-to-the-bottom-of-the-rod.md)
- [how the force between two charges can be re-imagined as a force mediated by the electric field](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-force-between-two-charges-can-be-re-imagined-as-a-force-mediated-by-the-electric-field.md)
- [the high conductivity within the carbon sheets by showing a constant flow of bright and fast-moving](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-high-conductivity-within-the-carbon-sheets-by-showing-a-constant-flow-of-bright-and-fast-moving.md)
- [compare the free precession of a disc and a prolate spheroid](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/compare-the-free-precession-of-a-disc-and-a-prolate-spheroid.md)
- [apply different transformations to a grid of particles representing a solid material and observe how](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/apply-different-transformations-to-a-grid-of-particles-representing-a-solid-material-and-observe-how.md)
- [visualize elasticity and Hooke's Law for an isotropic material](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-elasticity-and-hookes-law-for-an-isotropic-material.md)
- [Visualize how vector components change with the coordinate system](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-how-vector-components-change-with-the-coordinate-system.md)
- [how the difference in how an electric field propagates through a vacuum versus how it propagates thr](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-difference-in-how-an-electric-field-propagates-through-a-vacuum-versus-how-it-propagates-thr.md)
- [The polar coordinate tangent vectors depend on Cartesian basis vectors](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-polar-coordinate-tangent-vectors-depend-on-cartesian-basis-vectors.md)
- [the polarisation is proportional to the electric field within an isotropic medium that is contrary](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-polarisation-is-proportional-to-the-electric-field-within-an-isotropic-medium-that-is-contrary.md): the polarisation is proportional to the electric field within an isotropic medium that is contrary to non-isotropic medium
- [how the conductivity tensor affects the relationship between the electric field and current density](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-conductivity-tensor-affects-the-relationship-between-the-electric-field-and-current-density.md)
- [how tensors behave under a change of basis](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-tensors-behave-under-a-change-of-basis.md)
- [how a tensor's components change when you transform the coordinate system while the result of the co](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-a-tensors-components-change-when-you-transform-the-coordinate-system-while-the-result-of-the-co.md)
- [how the components of the tensor and two vectors change as the coordinate system rotates while the f](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-components-of-the-tensor-and-two-vectors-change-as-the-coordinate-system-rotates-while-the-f.md)
- [a type (2,0) tensor that is initially symmetric and the transformed tensor remains symmetric](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/a-type-2-0-tensor-that-is-initially-symmetric-and-the-transformed-tensor-remains-symmetric.md)
- [how the components of the tensor and two vectors change as the coordinate system rotates while the f](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-components-of-the-tensor-and-two-vectors-change-as-the-coordinate-system-rotates-while-the-1.md)
- [Show symmetric and anti-symmetric components of 2D tensor represented by a stretching ellipsoid and](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/show-symmetric-and-anti-symmetric-components-of-2d-tensor-represented-by-a-stretching-ellipsoid-and.md)
- [A tensor contraction only depends on its symmetric part if the contracted indices are symmetric in](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/a-tensor-contraction-only-depends-on-its-symmetric-part-if-the-contracted-indices-are-symmetric-in.md): A tensor contraction only depends on its symmetric part if the contracted indices are symmetric in the vectors
- [visualize a generic type (0, 3) tensor and then show how a zero tensor appears](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-a-generic-type-0-3-tensor-and-then-show-how-a-zero-tensor-appears.md)
- [visualize a general type (3,0) tensor as a 3D grid of spheres where each sphere corresponds to a com](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-a-general-type-3-0-tensor-as-a-3d-grid-of-spheres-where-each-sphere-corresponds-to-a-com.md)
- [a tensor as a physical object that exists independently of the coordinate system used to describe it](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/a-tensor-as-a-physical-object-that-exists-independently-of-the-coordinate-system-used-to-describe-it.md)
- [how the magnetic field vector points in the azimuthal direction and how its magnitude changes with d](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-magnetic-field-vector-points-in-the-azimuthal-direction-and-how-its-magnitude-changes-with-d.md)
- [the gradient of a scalar field is a covariant vector](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-gradient-of-a-scalar-field-is-a-covariant-vector.md)
- [the fictitious forces that arise in non-inertial frames of reference](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-fictitious-forces-that-arise-in-non-inertial-frames-of-reference.md)
- [how to calculate divergence in a curved coordinate system](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-to-calculate-divergence-in-a-curved-coordinate-system.md)
- [how the divergence field emerges from the antisymmetric tensor](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-divergence-field-emerges-from-the-antisymmetric-tensor.md)
- [the geometrical meaning of the Christoffel symbols in flat Euclidean space using the non-Cartesian s](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-geometrical-meaning-of-the-christoffel-symbols-in-flat-euclidean-space-using-the-non-cartesian-s.md)
- [Cylindrical has one scaled tangent direction while Spherical has two](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/cylindrical-has-one-scaled-tangent-direction-while-spherical-has-two.md)
- [Finding the Arc Length in Spherical Coordinates](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/finding-the-arc-length-in-spherical-coordinates.md)
- [visualize the underlying geometry and the tangent basis vectors defined by the metric demonstrating](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-the-underlying-geometry-and-the-tangent-basis-vectors-defined-by-the-metric-demonstrating.md)
- [display the instantaneous line element at a moving point and compare the standard Cartesian length c](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/display-the-instantaneous-line-element-at-a-moving-point-and-compare-the-standard-cartesian-length-c.md)
- [a 2D type (1,1) tensor under rotation under both linear transformation and non-linear transformation](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/a-2d-type-1-1-tensor-under-rotation-under-both-linear-transformation-and-non-linear-transformation.md)
- [the covariant derivative is indispensable in non-Cartesian or curved systems that distinguishes it f](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-covariant-derivative-is-indispensable-in-non-cartesian-or-curved-systems-that-distinguishes-it-f.md)
- [why the standard divergence formula requires spherical volume element](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/why-the-standard-divergence-formula-requires-spherical-volume-element.md)
- [the Laplace-Beltrami Operator measures the local curvature of the field by detecting peaks and troug](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-laplace-beltrami-operator-measures-the-local-curvature-of-the-field-by-detecting-peaks-and-troug.md)
- [the divergence of the tangent basis vectors illustrates why these coordinate systems have non-Cartes](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-divergence-of-the-tangent-basis-vectors-illustrates-why-these-coordinate-systems-have-non-cartes.md)
- [the square root of the determinant of the metric tensor unifies the Divergence of the Gradient in Cu](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-square-root-of-the-determinant-of-the-metric-tensor-unifies-the-divergence-of-the-gradient-in-cu.md)
- [the physical density is invariant as a geometric object but its coordinate representation changes](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-physical-density-is-invariant-as-a-geometric-object-but-its-coordinate-representation-changes.md)
- [Jacobian determinant for a composite coordinate transformation is the product of the individual Jaco](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/jacobian-determinant-for-a-composite-coordinate-transformation-is-the-product-of-the-individual-jaco.md)
- [the metric determinant scales vector operations](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-metric-determinant-scales-vector-operations.md)
- [the totally antisymmetric tensor in a flat two-dimensional space](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-totally-antisymmetric-tensor-in-a-flat-two-dimensional-space.md)
- [how two dynamic inputs determine the covariant components of the resulting vector](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-two-dynamic-inputs-determine-the-covariant-components-of-the-resulting-vector.md)
- [Visualizing the Curl of Dual Bases in Curvilinear Coordinates](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualizing-the-curl-of-dual-bases-in-curvilinear-coordinates.md)
- [What're geometric actions associated with antisymmetric tensors and symmetric tensors](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/whatre-geometric-actions-associated-with-antisymmetric-tensors-and-symmetric-tensors.md)
- [rigid-body motion using an orthogonal affine transformation](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/rigid-body-motion-using-an-orthogonal-affine-transformation.md)
- [visualize the density fields of Kinetic Energy Momentum and Angular Momentum as a function of time](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-the-density-fields-of-kinetic-energy-momentum-and-angular-momentum-as-a-function-of-time.md)
- [Visualize the area element (the Jacobian determinant) helps illustrate how the transformation stretc](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-the-area-element-the-jacobian-determinant-helps-illustrate-how-the-transformation-stretc.md)
- [Visualize how the Poisson's ratio approaches the incompressibility limit as the stiffness ratio incr](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-how-the-poissons-ratio-approaches-the-incompressibility-limit-as-the-stiffness-ratio-incr.md)
- [how the magnetic stress tensor decomposes to show that magnetic fields simultaneously exert tension](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-magnetic-stress-tensor-decomposes-to-show-that-magnetic-fields-simultaneously-exert-tension.md)
- [Visualize the electric field lines and the resulting surface forces between the attractive and repul](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-the-electric-field-lines-and-the-resulting-surface-forces-between-the-attractive-and-repul.md)
- [the local Lorentz force density is zero if there are no charges or currents present to act upon](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-local-lorentz-force-density-is-zero-if-there-are-no-charges-or-currents-present-to-act-upon.md)
- [stability and complexity of motion are governed by the relationship between the angular velocity and](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/stability-and-complexity-of-motion-are-governed-by-the-relationship-between-the-angular-velocity-and.md)
- [Centrifugal Force as the Stability Governor](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/centrifugal-force-as-the-stability-governor.md)
- [The centrifugal force depends only on the particle's perpendicular distance in a rotating reference](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-centrifugal-force-depends-only-on-the-particles-perpendicular-distance-in-a-rotating-reference.md)
- [Clarifying the Contributions of the Tidal Tensor Components](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/clarifying-the-contributions-of-the-tidal-tensor-components.md)
- [Visualizing the Magnetic Stress Tensor is highly illustrative of how magnetic fields exert forces](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualizing-the-magnetic-stress-tensor-is-highly-illustrative-of-how-magnetic-fields-exert-forces.md)
- [Visualize Electric Field Divergence from Charge and Magnetic Field Curl from Changing Electric Field](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-electric-field-divergence-from-charge-and-magnetic-field-curl-from-changing-electric-field.md)
- [Visualize the behavior of both intensive properties and extensive properties](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-the-behavior-of-both-intensive-properties-and-extensive-properties.md)
- [Visualize the conservation of momentum and the action-reaction forces through the two-body collision](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-the-conservation-of-momentum-and-the-action-reaction-forces-through-the-two-body-collision.md)
- [show the exponential decay over the time scale set by the characteristic charge relaxation time](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/show-the-exponential-decay-over-the-time-scale-set-by-the-characteristic-charge-relaxation-time.md)
- [Only the component of the current density vector that is exactly perpendicular to the surface contri](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/only-the-component-of-the-current-density-vector-that-is-exactly-perpendicular-to-the-surface-contri.md): Only the component of the current density vector that is exactly perpendicular to the surface contributes to the flux
- [the difference between concentration change due to external flow and concentration change due to int](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-difference-between-concentration-change-due-to-external-flow-and-concentration-change-due-to-int.md): the difference between concentration change due to external flow and concentration change due to internal processes
- [water exits a faucet and accelerates under gravity while its velocity increases as it falls](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/water-exits-a-faucet-and-accelerates-under-gravity-while-its-velocity-increases-as-it-falls.md)
- [how the shape and peak height change for pure diffusion and pure decay and the combined scenario](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-shape-and-peak-height-change-for-pure-diffusion-and-pure-decay-and-the-combined-scenario.md)
- [Point Source Diffusion-From Transient Pulse to Steady Source](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/point-source-diffusion-from-transient-pulse-to-steady-source.md)
- [Electrostatic Potentials-Numerical Validation of Point vs Distributed Charge](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/electrostatic-potentials-numerical-validation-of-point-vs-distributed-charge.md)
- [How the delta function is used to model charge distributions concentrated on a line or a surface ins](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-delta-function-is-used-to-model-charge-distributions-concentrated-on-a-line-or-a-surface-ins.md)
- [the concentration profile over time for three scenario-pure diffusion and pure convection and the co](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-concentration-profile-over-time-for-three-scenario-pure-diffusion-and-pure-convection-and-the-co.md): the concentration profile over time for three scenario-pure diffusion and pure convection and the combined case
- [Fick's second law is used to Chemical Mixing and Heat Transfer and Semiconductor Doping](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/ficks-second-law-is-used-to-chemical-mixing-and-heat-transfer-and-semiconductor-doping.md)
- [Dissecting the Deterministic Roles of Diffusivity Decay and Dimensionality in PDEs](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/dissecting-the-deterministic-roles-of-diffusivity-decay-and-dimensionality-in-pdes.md)
- [Core Scientific Laws and Thermodynamic Properties Illustrated Through Dynamic Visualization](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/core-scientific-laws-and-thermodynamic-properties-illustrated-through-dynamic-visualization.md)
- [visualize how an initial wave profile splits into two equal and opposite-traveling components](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-how-an-initial-wave-profile-splits-into-two-equal-and-opposite-traveling-components.md)
- [visually compares the behavior of an undamped wave and a damped wave over time](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visually-compares-the-behavior-of-an-undamped-wave-and-a-damped-wave-over-time.md)
- [visualize the wave equation solution for the condition where the string starts with zero initial dis](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-the-wave-equation-solution-for-the-condition-where-the-string-starts-with-zero-initial-dis.md): visualize the wave equation solution for the condition where the string starts with zero initial displacement but is given a velocity profile
- [the net forces and tension acting on a small element of a vibrating membrane](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-net-forces-and-tension-acting-on-a-small-element-of-a-vibrating-membrane.md)
- [Dynamic Visualization of Wave Equation Principles-Analyzing Force Balance and Traveling Wave Compone](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/dynamic-visualization-of-wave-equation-principles-analyzing-force-balance-and-traveling-wave-compone.md): Dynamic Visualization of Wave Equation Principles-Analyzing Force Balance and Traveling Wave Components and the Effects of Damping
- [Boundary conditions define the allowed solutions (eigenmodes) and the natural frequencies](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/boundary-conditions-define-the-allowed-solutions-eigenmodes-and-the-natural-frequencies.md): Boundary conditions define the allowed solutions (eigenmodes) and the natural frequencies (eigenvalues)
- [The demonstration of the nature of transient heat diffusion and the importance of the Fourier number](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-demonstration-of-the-nature-of-transient-heat-diffusion-and-the-importance-of-the-fourier-number.md)
- [how the Neumann boundary condition dictates the behavior of a sealed system](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-neumann-boundary-condition-dictates-the-behavior-of-a-sealed-system.md)
- [using the Finite Difference Method to solve the 1D wave equation with the mass-loaded boundary condi](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/using-the-finite-difference-method-to-solve-the-1d-wave-equation-with-the-mass-loaded-boundary-condi.md): using the Finite Difference Method to solve the 1D wave equation with the mass-loaded boundary condition
- [Steady-State Heat Transfer-Comparison of Dirichlet and Robin (Newton's Cooling) and Neumann Boundary](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/steady-state-heat-transfer-comparison-of-dirichlet-and-robin-newtons-cooling-and-neumann-boundary.md): Steady-State Heat Transfer-Comparison of Dirichlet and Robin (Newton's Cooling) and Neumann Boundary Conditions
- [how the choice of boundary condition fundamentally dictates the long-term equilibrium and the result](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-choice-of-boundary-condition-fundamentally-dictates-the-long-term-equilibrium-and-the-result.md): how the choice of boundary condition fundamentally dictates the long-term equilibrium and the resulting flow (flux) across the boundary of a material
- [The initial condition is the crucial starting point for any time-dependent simulation](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-initial-condition-is-the-crucial-starting-point-for-any-time-dependent-simulation.md)
- [Visualize how the string vibrates over time as a superposition of standing waves](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-how-the-string-vibrates-over-time-as-a-superposition-of-standing-waves.md)
- [Visualize solutions to Poisson's Equation and Laplace's Equation](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-solutions-to-poissons-equation-and-laplaces-equation.md)
- [how the string's equilibrium is fundamentally shifted by the constant external force](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-strings-equilibrium-is-fundamentally-shifted-by-the-constant-external-force.md)
- [the stationary solution to Poisson's equation driven by a Dirac delta function source](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-stationary-solution-to-poissons-equation-driven-by-a-dirac-delta-function-source.md)
- [how the string's equilibrium is fundamentally shifted by the constant external force](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-strings-equilibrium-is-fundamentally-shifted-by-the-constant-external-force-1.md)
- [the stationary solution to Poisson's equation driven by a Dirac delta function source](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-stationary-solution-to-poissons-equation-driven-by-a-dirac-delta-function-source-1.md)
- [the analytical solution for a specific mode of the Helmholtz equation using Bessel functions](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-analytical-solution-for-a-specific-mode-of-the-helmholtz-equation-using-bessel-functions.md)
- [Plot the region where the Principal Symbol is zero for both the Wave Operator-Hyperbolic and the Dif](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/plot-the-region-where-the-principal-symbol-is-zero-for-both-the-wave-operator-hyperbolic-and-the-dif.md): Plot the region where the Principal Symbol is zero for both the Wave Operator-Hyperbolic and the Diffusion Operator-Parabolic
- [Stability Analysis of Reaction-Diffusion Equations-Linearization Demonstrating Growth and Decay](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/stability-analysis-of-reaction-diffusion-equations-linearization-demonstrating-growth-and-decay.md)
- [the convective transport of momentum through a surface element by a fluid moving with velocity and 5](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-convective-transport-of-momentum-through-a-surface-element-by-a-fluid-moving-with-velocity-and-5.md): the convective transport of momentum through a surface element by a fluid moving with velocity and 5 examples
- [From Dust to D-Force-Visualizing the Cauchy Momentum Equation](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/from-dust-to-d-force-visualizing-the-cauchy-momentum-equation.md)
- [Visualize hydrostatic equilibrium is best done by focusing on the balance of forces acting on an inf](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-hydrostatic-equilibrium-is-best-done-by-focusing-on-the-balance-of-forces-acting-on-an-inf.md): Visualize hydrostatic equilibrium is best done by focusing on the balance of forces acting on an infinitesimal volume of fluid and the resulting pressure distribution
- [visualize the inverse relationship between fluid speed and pressure along a streamline-Bernoulli's p](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualize-the-inverse-relationship-between-fluid-speed-and-pressure-along-a-streamline-bernoullis-p.md): visualize the inverse relationship between fluid speed and pressure along a streamline-Bernoulli's principle
- [how pressure and temperature and density and velocity change as a gas flows isentropically through a](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-pressure-and-temperature-and-density-and-velocity-change-as-a-gas-flows-isentropically-through-a.md): how pressure and temperature and density and velocity change as a gas flows isentropically through a convergent-divergent nozzle
- [Hagen-Poiseuille flow-or called Poiseuille flow through a circular pipe](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/hagen-poiseuille-flow-or-called-poiseuille-flow-through-a-circular-pipe.md)
- [the relationship between numerical modeling and analytical solutions-Poiseuille's Law in fluid mecha](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/the-relationship-between-numerical-modeling-and-analytical-solutions-poiseuilles-law-in-fluid-mecha.md): the relationship between numerical modeling and analytical solutions-Poiseuille's Law in fluid mechanics
- [how the three components—the quasi-static response and the transient response and the steady-state f](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-three-components-the-quasi-static-response-and-the-transient-response-and-the-steady-state-f.md): how the three components—the quasi-static response and the transient response and the steady-state forced response combine to form the total solution
- [superposition principle in both electrostatics and wave propagation](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/superposition-principle-in-both-electrostatics-and-wave-propagation.md)
- [how the three components—the quasi-static response and the transient response and the steady-state f](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-three-components-the-quasi-static-response-and-the-transient-response-and-the-steady-state-1.md): how the three components—the quasi-static response and the transient response and the steady-state forced response combine to form the total solution
- [Heat Conduction with Inhomogeneous Condition and Homogeneous Condition](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/heat-conduction-with-inhomogeneous-condition-and-homogeneous-condition.md)
- [From Indices to Inertia-Visualizing Rotation via Tensor Mechanics (IIR-TM)](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/from-indices-to-inertia-visualizing-rotation-via-tensor-mechanics-iir-tm.md)
- [Vector Triple Product-From Geometry to Efficiency](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/vector-triple-product-from-geometry-to-efficiency.md)
- [Comparative Analysis of Dimensional Scaling](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/comparative-analysis-of-dimensional-scaling.md)
- [Analysis of Vector Field Dynamics-Position vs. Gravitation](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/analysis-of-vector-field-dynamics-position-vs.-gravitation.md)
- [Algebraic Cross Product vs. Geometric Lie Bracket](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/algebraic-cross-product-vs.-geometric-lie-bracket.md)
- [Parallelogram Diagonals Orthogonality Demo](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/parallelogram-diagonals-orthogonality-demo.md)
- [Geometric Analysis of Diagonal Angles](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/geometric-analysis-of-diagonal-angles.md)
- [Why the difference vector is orthogonal at the points of closest approach](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/why-the-difference-vector-is-orthogonal-at-the-points-of-closest-approach.md)
- [How the relationship between the cross product and the geometry of two vectors applies in physics to](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/how-the-relationship-between-the-cross-product-and-the-geometry-of-two-vectors-applies-in-physics-to.md): How the relationship between the cross product and the geometry of two vectors applies in physics to calculate torque or magnetic force
- [use a simple Euler numerical integration method to simulate the precession over time](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/use-a-simple-euler-numerical-integration-method-to-simulate-the-precession-over-time.md)
- [Display a 3D plot of the helical path to emphasize the circular motion](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/display-a-3d-plot-of-the-helical-path-to-emphasize-the-circular-motion.md)
- [Visualization of Three Vector Fields with Different Divergence and Curl (TVF-DC)](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualization-of-three-vector-fields-with-different-divergence-and-curl-tvf-dc.md)
- [Visualizing the Geometric Algebra of Differential Identities (GA-DI)](/all/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/visualizing-the-geometric-algebra-of-differential-identities-ga-di.md)