🧄Proof and Derivation
🧄Proving the Cross Product Rules with the Levi-Civita Symbol🧄Proving the Epsilon-Delta Relation and the Bac-Cab Rule🧄Simplifying Levi-Civita and Kronecker Delta Identities🧄Dot Cross and Triple Products🧄Why a Cube's Diagonal Angle Never Changes🧄How the Cross Product Relates to the Sine of an Angle🧄Finding the Shortest Distance and Proving Orthogonality for Skew Lines🧄A Study of Helical Trajectories and Vector Dynamics🧄The Power of Cross Products: A Visual Guide to Precessing Vectors🧄Divergence and Curl Analysis of Vector Fields🧄Unpacking Vector Identities: How to Apply Divergence and Curl Rules🧄Commutativity and Anti-symmetry in Vector Calculus Identities🧄Double Curl Identity Proof using the epsilon-delta Relation🧄The Orthogonality of the Cross Product Proved by the Levi-Civita Symbol and Index Notation🧄Surface Parametrisation and the Verification of the Gradient-Normal Relationship🧄Proof and Implications of a Vector Operator Identity🧄Conditions for a Scalar Field Identity🧄Solution and Proof for a Vector Identity and Divergence Problem🧄Kinematics and Vector Calculus of a Rotating Rigid Body🧄Work Done by a Non-Conservative Force and Conservative Force🧄The Lorentz Force and the Principle of Zero Work Done by a Magnetic Field🧄Calculating the Area of a Half-Sphere Using Cylindrical Coordinates🧄Divergence Theorem Analysis of a Vector Field with Power-Law Components🧄Total Mass in a Cube vs. a Sphere🧄Momentum of a Divergence-Free Fluid in a Cubic Domain🧄Total Mass Flux Through Cylindrical Surfaces🧄Analysis of Forces and Torques on a Current Loop in a Uniform Magnetic Field🧄Computing the Integral of a Static Electromagnetic Field🧄Surface Integral to Volume Integral Conversion Using the Divergence Theorem🧄Circulation Integral vs. Surface Integral🧄Using Stokes' Theorem with a Constant Scalar Field🧄Verification of the Divergence Theorem for a Rotating Fluid Flow🧄Integral of a Curl-Free Vector Field🧄Boundary-Driven Cancellation in Vector Field Integrals🧄The Vanishing Curl Integral🧄Proving the Generalized Curl Theorem🧄Computing the Magnetic Field and its Curl from a Dipole Vector Potential🧄Proving Contravariant Vector Components Using the Dual Basis🧄Verification of Orthogonal Tangent Vector Bases in Cylindrical and Spherical Coordinates🧄Vector Field Analysis in Cylindrical Coordinates🧄Vector Field Singularities and Stokes' Theorem🧄Compute Parabolic coordinates-related properties🧄Analyze Flux and Laplacian of The Yukawa Potential🧄Verification of Vector Calculus Identities in Different Coordinate Systems🧄Analysis of a Divergence-Free Vector Field🧄The Uniqueness Theorem for Vector Fields🧄Analysis of Electric Dipole Force Field🧄Derivation of Tensor Transformation Properties for Mixed Tensors🧄The Polar Tensor Basis in Cartesian Form🧄Verifying the Rank Two Zero Tensor🧄Tensor Analysis of Electric Susceptibility in Anisotropic Media🧄Analysis of Ohm's Law in an Anisotropic Medium🧄Verifying Tensor Transformations🧄Proof of Coordinate Independence of Tensor Contraction🧄Proof of a Tensor's Invariance Property🧄Proving Symmetry of a Rank-2 Tensor🧄Tensor Symmetrization and Anti-Symmetrization Properties🧄Symmetric and Antisymmetric Tensor Contractions🧄The Uniqueness of the Zero Tensor under Specific Symmetry Constraints🧄Counting Independent Tensor Components Based on Symmetry🧄Transformation of the Inverse Metric Tensor🧄Finding the Covariant Components of a Magnetic Field🧄Covariant Nature of the Gradient🧄Christoffel Symbol Transformation Rule Derivation🧄Contraction of the Christoffel Symbols and the Metric Determinant🧄Divergence of an Antisymmetric Tensor in Terms of the Metric Determinant🧄Calculation of the Metric Tensor and Christoffel Symbols in Spherical Coordinates🧄Christoffel Symbols for Cylindrical Coordinates🧄Finding Arc Length and Curve Length in Spherical Coordinates🧄Solving for Metric Tensors and Christoffel Symbols🧄Metric Tensor and Line Element in Non-Orthogonal Coordinates🧄Tensor vs. Non-Tensor Transformation of Derivatives🧄Verification of Covariant Derivative Identities🧄Divergence in Spherical Coordinates Derivation and Verification🧄Laplace Operator Derivation and Verification in Cylindrical Coordinates🧄Divergence of Tangent Basis Vectors in Curvilinear Coordinates🧄Derivation of the Laplacian Operator in General Curvilinear Coordinates🧄Verification of Tensor Density Operations🧄Verification of the Product Rule for Jacobian Determinants and Tensor Density Transformation🧄Metric Determinant and Cross Product in Scaled Coordinates🧄Vanishing Divergence of the Levi-Civita Tensor🧄Curl and Vector Cross-Product Identity in General Coordinates🧄Curl of the Dual Basis in Cylindrical and Spherical Coordinates🧄Proof of Covariant Index Anti-Symmetrisation🧄Affine Transformations and the Orthogonality of Cartesian Rotations🧄Fluid Mechanics Integrals for Mass and Motion🧄Volume Elements in Non-Cartesian Coordinates (Jacobian Method)🧄Young's Modulus and Poisson's Ratio in Terms of Bulk and Shear Moduli🧄Tensor Analysis of the Magnetic Stress Tensor🧄Surface Force for Two Equal Charges🧄Total Electromagnetic Force in a Source-Free Static Volume🧄Proof of the Rotational Identity🧄Finding the Generalized Inertia Tensor for the Coupled Mass System🧄Tensor Form of the Centrifugal Force in Rotating Frames🧄Derivation and Calculation of the Gravitational Tidal Tensor🧄Conversion of Total Magnetic Force to a Surface Integral via the Maxwell Stress Tensor🧄Verifying the Inhomogeneous Maxwell's Equations in Spacetime
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