Cartesian Tensors. C54H -Astrophysical Fluid Dynamics. 1. Cartesian Tensors. Reference: Jeffreys Cartesian Tensors. 1 Coordinates and Vectors. Coordinates. Download Citation on ResearchGate | Cartesian tensors / by Harold Jeffreys | Incluye índice }. Harold Jeffreys-Cartesian Tensors -Cambridge University Press ().pdf – Download as PDF File .pdf) or read online.
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In geometry and linear algebraa Cartesian tensor uses an orthonormal basis to represent a tensor jeffeys 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 acrtesian, any finite-dimensional vector space over the field of real numbers that has an inner product.
Use of Cartesian tensors occurs in physics and engineeringsuch 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 mechanicsor even necessary, as in general relativity.
While orthonormal bases may be found for some such coordinate systems e. The transformation is a passive transformationsince the coordinates are changed and not the physical system. Each basis vector points along the x- y- and z-axes, and the vectors are all unit vectors or normalizedso the basis is orthonormal.
Throughout, when referring to Cartesian coordinates in three dimensionsa right-handed system is assumed and this is much more common than a left-handed system in practice, see orientation vector space for details. It is common and helpful to display the basis vectors as column vectors.
A row ejffreys 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. A more general notation is tensor index notationwhich gensors the flexibility of numerical values rather than fixed coordinate labels. In general, the notation e 1e 2e 3 refers to any basis, and A 1A 2A 3 refers to the corresponding coordinate system; although here they are restricted to the Cartesian system.
It is standard to ccartesian the Einstein notation —the summation sign for summation over an index that is present exactly twice within a term may be suppressed for notational conciseness:. An advantage of the index notation over coordinate-specific notations is jeffryes independence of the dimension of the underlying vector space, i. Previously, the Cartesian labels x, y, z were just labels cartesiam not indices. See matrix multiplication for the notational correspondence between matrices and the jedfreys and tensor products.
More generally, whether or not T is a tensor product of two vectors, it is always a linear combination of the basis tensors with coordinates T xxT xySecond order tensors occur naturally in physics and engineering when physical quantities have directional dependence in the system, often in a “stimulus-response” way. This can be mathematically seen through one aspect of tensors – they are multilinear functions.
A second order tensor T which takes in a vector u of some magnitude and direction will return a vector v ; of a different magnitude and in a different direction to uin general. The function, matrix, and index notations all mean the same thing. The matrix forms provide a clear display of the components, while the index form allows easier tensor-algebraic manipulation of the formulae in a compact manner.
Both provide the physical interpretation of directions ; jeffryes have one direction, while second order tensors connect two directions together. One can associate a tensor index or coordinate label with a basis vector direction.
The use of second order tensors are the minimum to describe changes in magnitudes and directions of vectors, as the dot product of two vectors is always a scalar, while the cross product of two vectors is always a pseudovector perpendicular to the plane defined by the vectors, so these products of vectors alone cannot jefreys a new vector of any magnitude in any direction.
See also below for more on the dot and cross products. The tensor product of two vectors is a second order tensor, although this has no obvious directional interpretation by itself. The previous idea can be continued: The tensor T is linear caartesian both input vectors. For the above cases: As multilinear maps for further generalizations and details.
The concepts above also apply to pseudovectors in the same way as for vectors. The vectors and tensors themselves can vary jefreys throughout space, in which case we have vector fields and tensor fieldsand can also depend on time. See also constitutive equation for more tenssors examples.
Each basis vector e i points along tensorrs positive x i axis, with the basis being orthonormal. Component j of e i is given by the Kronecker delta:. Consider the case of rectangular coordinate systems with orthonormal bases only. It is possible to have a coordinate system with etnsors geometry if the basis vectors are all mutually perpendicular and not normalized, in which case the basis is ortho gonal but not ortho normal.
However, orthonormal tenssors are easier to manipulate and are often used in practice. The following results are true for orthonormal bases, not orthogonal ones.
In one rectangular coordinate system, x as a cartesoan has coordinates x i and basis vectors e iwhile as a covector it has coordinates x i and basis covectors e iand we have:. In another rectangular coordinate system, x as a contravector has coordinates x i and bases e iwhile as a covector it has coordinates x i and bases e iand we have:.
Each new coordinate is a function of all cartesiaj old ones, and vice versa for the inverse function:. The difference between each of these transformations is shown conventionally through the indices as superscripts for contravariance and subscripts for covariance, and the coordinates and bases are linearly transformed according to the following rules:.
If L is an orthogonal transformation orthogonal matrixthe objects transforming by it are defined as Cartesian tensors. This geometrically has the interpretation that a rectangular coordinate system is mapped to another rectangular coordinate system, in meffreys the norm of the vector x is preserved and distances are preserved. There are considerable algebraic simplifications, the matrix transpose is the inverse from the definition of an orthogonal transformation:.
From the previous table, orthogonal transformations of covectors and contravectors are identical. There is no need to differ between raising and lowering indicesand in this context and applications to physics and engineering the indices are usually all subscripted to remove confusion for exponents. All indices will be lowered in the remainder of this article. One can determine the actual gensors and lowered indices by considering which quantities are covectors or contravectors, and the relevant transformation rules.
Exactly the same transformation rules apply to any vector anot only the position vector. If its components a i do not transform according to the rules, a is not a vector.
In the change of coordinates, L is a matrixused to relate two rectangular coordinate systems with orthonormal bases together. For the tensor relating a vector to a vector, the vectors and tensors throughout the equation all belong to the same coordinate system and basis. The entries of L are partial derivatives of the new or old coordinates with respect to the old or new coordinates, respectively.
Differentiating x i with respect to x tensore. There is a partially mnemonical correspondence between tnsors positions attached to L and in the partial derivative: Conversely, differentiating x j with respect to x i:. As with jeffrrys linear transformations, L depends on the basis chosen.
For two orthonormal bases. Hence the components reduce to direction cosines between the x i and x j axes:. The geometric interpretation is the x i components equal to the sum of projecting the x j components onto the x j axes.
Therefore, while the L matrices are still orthogonal, they are not symmetric. Apart from a rotation about any one axis, in which the x i and x i for some i coincide, the angles are not the same as Euler anglesand so the L matrices are not the same as the rotation matrices. The dot product and cross product occur very frequently, in applications of vector analysis to physics and engineering, examples include:.
For jeffryes pairs we have. Replacing Tensprs labels by index notation as shown abovethese results can be summarized by. In addition, each metric tensor component g ij with respect to any basis is the dot product of a pairing of basis vectors:.
This is not true for general bases: This also applies more generally to any coordinate systems, not just rectangular ones; the dot product in one coordinate system is the same in any other. Again, assuming a right-handed 3d Cartesian coordinate system, cyclic permutations in perpendicular directions yield the next vector in the cyclic collection of vectors:.
Cartesian tensors – Sir Harold Jeffreys – Google Books
These permutation relations and their corresponding values are important, and there is an object jeffteys with this property: The Levi-Civita cartewian entries can be represented by the Cartesian ccartesian. The scalar triple product can now be tenslrs.
This in turn can be used to rewrite the cross product of two vectors as follows:. Contrary to its appearance, the Levi-Civita symbol is not a tensorbut a pseudotensorthe components transform according to:. The tensor index notation applies to any object which has entities that form multidimensional arrays — not everything with indices is a tensor by default.
Instead, tensors are defined by caartesian their coordinates and basis elements change under a transformation from one coordinate system to another. Note the cross product of two vectors is a pseudovector, while the cross product of a pseudovector with a vector is another vector. The index forms of the dot and cross products, together with this identity, greatly facilitate the manipulation and derivation of other identities in vector calculus and algebra, which in turn are cartesiam extensively in physics and engineering.
For instance, it is clear the dot and cross products are distributive over vector addition:. Although the procedure is less obvious, the vector triple product can also be derived.
Rewriting in index notation:. Note this is antisymmetric in b and cas expected from the left hand side. Similarly, via index notation or even just cyclically relabelling aband c in the previous result and taking the negative:.
More complex identities, like quadruple products. Tensors are defined as quantities which transform in a certain way under linear transformations of coordinates. If R does not transform according to this rule – whatever quantity R may be, it’s not an order 2 tensor.
Cartesian Tensors – Harold Jeffreys – Google Books
For a pseudotensor S of order pthe components transform according to. The antisymmetric nature of the cross product can be recast into a tensorial form as follows. As the cross product is linear in a and bthe components of T can be found by inspection, and they are:.
This transforms as a tensor, not a pseudotensor.
For an example in electromagnetismwhile the electric field E is a vector fieldthe magnetic field B is a pseudovector field. These fields are defined from the Lorentz force for a particle of electric charge q traveling at velocity v:. Cartsian a pseudovector is explicitly given jeffrsys a cross product of two vectors as opposed to entering the cross product with another vectorthen such pseudovectors can also be written as antisymmetric tensors of second order, with each entry a component of the cross product.
Although Cartesian tensors do not occur in the theory of relativity; the tensor form of orbital angular momentum J enters the spacelike part of the relativistic angular momentum tensor, and the above jefreys form of the magnetic field B enters the spacelike part of the electromagnetic tensor.