Cartesian tensor

In geometry and linear algebra, a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components from one such basis to another is through an orthogonal transformation. x ¯ j = x i ∂ x ¯ j ∂ x i ↿⇂ x j = x ¯ i ∂ x j ∂ x ¯ i {displaystyle {egin{array}{c}{ar {x}}_{j}=x_{i}{frac {partial {ar {x}}_{j}}{partial x_{i}}}\upharpoonleft downharpoonright \x_{j}={ar {x}}_{i}{frac {partial x_{j}}{partial {ar {x}}_{i}}}end{array}}} x ¯ j = x i ( e ¯ i ⋅ e j ) = x i cos ⁡ θ i j ↿⇂ x j = x ¯ i ( e i ⋅ e ¯ j ) = x ¯ i cos ⁡ θ j i {displaystyle {egin{array}{c}{ar {x}}_{j}=x_{i}left({ar {mathbf {e} }}_{i}cdot mathbf {e} _{j} ight)=x_{i}cos heta _{ij}\upharpoonleft downharpoonright \x_{j}={ar {x}}_{i}left(mathbf {e} _{i}cdot {ar {mathbf {e} }}_{j} ight)={ar {x}}_{i}cos heta _{ji}end{array}}} In geometry and linear algebra, a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components from one such basis to another is through an orthogonal transformation. The most familiar coordinate systems are the two-dimensional and three-dimensional Cartesian coordinate systems. Cartesian tensors may be used with any Euclidean space, or more technically, any finite-dimensional vector space over the field of real numbers that has an inner product. Use of Cartesian tensors occurs in physics and engineering, such as with the Cauchy stress tensor and the moment of inertia tensor in rigid body dynamics. Sometimes general curvilinear coordinates are convenient, as in high-deformation continuum mechanics, or even necessary, as in general relativity. While orthonormal bases may be found for some such coordinate systems (e.g. tangent to spherical coordinates), Cartesian tensors may provide considerable simplification for applications in which rotations of rectilinear coordinate axes suffice. The transformation is a passive transformation, since the coordinates are changed and not the physical system. In 3d Euclidean space, ℝ3, the standard basis is ex, ey, ez. Each basis vector points along the x-, y-, and z-axes, and the vectors are all unit vectors (or normalized), so the basis is orthonormal. Throughout, when referring to Cartesian coordinates in three dimensions, a right-handed system is assumed and this is much more common than a left-handed system in practice, see orientation (vector space) for details.simiFor Cartesian tensors of order 1, a Cartesian vector a can be written algebraically as a linear combination of the basis vectors ex, ey, ez: where the coordinates of the vector with respect to the Cartesian basis are denoted ax, ay, az. It is common and helpful to display the basis vectors as column vectors when we have a coordinate vector in a column vector representation: A row vector representation is also legitimate, although in the context of general curvilinear coordinate systems the row and column vector representations are used separately for specific reasons – see Einstein notation and covariance and contravariance of vectors for why.