Tensor calculus

Tensor calculus is the study of tensor algebra, tensor fields, and their derivatives in coordinate-independent form. It provides tools for calculations on curved spaces and in changing coordinates.

Tensor: A multilinear object whose components transform according to a prescribed rule under changes of basis. Tensor calculus applies algebraic and differential operations to these objects.
Covariant derivative: A derivative on tensor fields that accounts for changes in the underlying geometry or basis. It extends ordinary differentiation while preserving tensorial transformation behavior.
General relativity: Einstein's theory of gravitation, in which spacetime geometry is dynamical and matter shapes curvature. Its field equations relate spacetime curvature tensors to matter and energy.
Partial derivative: The derivative of a function with respect to one coordinate while other coordinates are held fixed. Unlike a covariant derivative, it does not generally transform as a tensor when applied to tensor components.
Einstein notation: A convention in which repeated indices imply summation over their range. It makes tensor equations compact while keeping index contractions explicit.
Affine connection: A structure that specifies how to differentiate vector fields and compare tangent vectors at nearby points. Its coefficients supply the correction terms in covariant derivatives.
Continuum mechanics: The study of the deformation and motion of materials treated as continuous matter. Stress, strain, and deformation are represented with tensor fields.
Lie derivative: A derivative measuring how a tensor field changes along the flow generated by a vector field. It compares fields through a flow rather than through a chosen affine connection.
Coordinate system: A method for assigning numerical coordinates to points in a space. Tensor components depend on coordinates even when the geometric object does not.
Christoffel symbols: Coordinate-dependent coefficients that represent an affine connection in a chosen coordinate system. They make covariant-derivative calculations explicit, though they are not tensor components.

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