Tensors are mathematical objects that generalize scalars, vectors, and matrices, providing a powerful framework for describing physical quantities independently of the coordinate system used. Under a rotation of axes, tensor components transform according to specific rules that preserve the underlying physical relationships. Tensor operations include contraction, which reduces the order of a tensor by summing over a pair of indices, and the quotient rule, which provides a criterion for identifying tensors from their relationships with other tensorial quantities. The Kronecker delta serves as an identity tensor, while the Levi-Civita symbol is essential in expressing cross products and other antisymmetric operations. Pseudotensors differ from ordinary tensors in their transformation properties under reflections, and pseudovectors, also called axial vectors, transform differently from polar vectors under improper rotations. Curvilinear coordinates, such as cylindrical and spherical coordinates, require special attention because the coordinate basis vectors and their scale factors may vary with position. Vector operations in curvilinear coordinates, including the gradient, divergence, curl, and Laplacian, must therefore account for the geometry of the coordinate system. The metric tensor describes distances, angles, and the geometric relationships between coordinate directions, while also providing the means to raise and lower tensor indices. In classical mechanics, the moment of inertia is represented by a second-order tensor that relates angular velocity to angular momentum in three-dimensional motion. In continuum mechanics, the stress tensor describes internal forces acting across surfaces, while in electromagnetism, the electric susceptibility tensor characterizes how a material’s polarization responds to an applied electric field. Together, these concepts make tensor analysis indispensable in physics, engineering, differential geometry, and the mathematical description of systems whose properties must remain consistent under changes of coordinates.