Concept

Covariance and contravariance of vectors

Related lectures (31)

Covers linear pressure vessels and the basics of differential geometry of surfaces, including covariant and contravariant base vectors.

Shells I: Mechanics of Slender Structures

Covers linear and membrane theories of pressure vessels, differential geometry of surfaces, and the reduction of dimensionality from 3D to 2D.

Covariant and Contravariant Tensors: Maxwell Field Strength

Explores contravariant and covariant tensors, Maxwell field strength, and Lorentz invariance in linear transformations and metric properties.

Linear Shell Theory: Equilibrium and Energy

Covers the expression of the Kirchhoff-Saint Venant energy in a covariant setting and explores equilibrium equations for spherical shells and linear shell theory.

Mechanics of Slender Structure

Covers the expression of the Kirchhoff-St. Venant energy in a covariant setting and the equilibrium equations for spherical shells, among other topics.

Lorentz Transformations and Covariant Tensors

Explores Lorentz transformations, covariant tensors, rotational invariance, and linear transformations in vector spaces.

Differential Geometry of Surfaces

Covers the fundamentals of differential geometry of surfaces, including the equilibrium of shells, pressure vessels, and the curvature of surfaces.

Tensors & Indices Recap

Covers the basics of tensors and indices, including contravariant vectors and tensor operations.

Special Relativity

Covers the main points of special relativity, including symmetries, transformations, 4-vectors, Maxwell equations, and proper time.

Rotating Vectors: Fish Formula

Covers the interpretation of rotation in a coordinate system and the fish formula.

Lorentz Invariance and Covariant Tensors

Explores Lorentz invariance, tensors in vector spaces, and electromagnetic potentials.

Tensors and Lorentz Transformations

Covers tensors, Lorentz transformations, and electrodynamics in relativistic notation.

Linear Shell Theory in Mechanics of Slender Structures

Explores linear shell theory in analyzing spherical shell behavior under pressure loading and membrane forces.

Linear Independence and Bases in Vector Spaces

Explains linear independence, bases, and dimension in vector spaces, including the importance of the order of vectors in a basis.

Covariance and Contravariance of Vectors

Explains covariance and contravariance of vectors in multilinear algebra and tensor analysis, focusing on their behavior under changes in basis and scale.

Eigenvalues and Eigenvectors in 3D

Explores eigenvalues and eigenvectors in 3D linear algebra, covering characteristic polynomials, stability under transformations, and real roots.

Membrane Theory of Shells: Cylindrical and Spherical Structures

Analyzes membrane forces in thin shell structures using the membrane theory of shells.

Covariant Derivatives and Christoffel Symbols

Covers accelerated and inertial coordinate systems, Jacobian, volume elements, covariant derivatives, Christoffel symbols, Lorentz case, and metric tensor properties.

Orthogonality and Least Squares

Introduces orthogonality between vectors, angles, and orthogonal complement properties in vector spaces.

Shells I

Covers linear pressure vessels, thin shells, and critical buckling pressure, emphasizing the dimensional reduction from 3D to 2D.