Contravariant basis vectors

For each coordinate 2 in the full space, we may define a covariant basis vector 0R /02 and a contravariant basis vector 02 /0R, which obey orthogonality and completeness relations... 

A generalized set of reciprocal vectors for a constrained system is defined here to be any set off contravariant basis vectors b, ..., b- and K covariant basis... 

Given a point on the interface r — S z, 9), there is a tangent plane passing through it. The contravariant basis vectors for the tangent plane are given by... 

This may be confirmed by expanding an arbitary contravariant Cartesian vector (with a raised bead index) in a basis of a and m vectors and confirming that one recovers the original vector if such an expansion vector is left-multiplied by the RHS of Eq. (2.149). 

If the set 3N contravariant Cartesian vectors given by the / a vectors and K m vectors form a complete basis for 3N space of Cartesian vectors, which we will hereafter assume to be true, then they must also obey a completeness relation ... 

To determine the matrix, let us introduce the basis vectors and ey with respect to the plane throngh the vectors n,. and. The vector is perpendicnlar to the reference planes, whereas the vector Cy is parallel to them. Transforming these vectors into spherical basis vectors yields the contravariant spherical basis vectors [30] e"(nj ) and e (n J, which are rotated with respect to the vectors e (n,) and (n ) through the angles

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On the other hand, the reaction rate Jr r = 1/t) may be obtained by multiplying (4.43) with vectors of contravariant basis gr (see (A.89)). Inserting in such product from (4.33), from the relation between contra- and covariant bases in V (see (A.86)) and from (4.40), we obtain (by using of orthonormality of ) the relation between rates (reversal to (4.44))... 

The expression (18) features in the calculation of surface gradients. (An alternative derivation for the normal is also available through n = ti x t2.) It is useful to introduce the covariant basis vectors and in terms of the contravariant ones as follows ... 

The two transformations in (10.2.2) are said to be contragredient, the first (typical of basis vectors) being covariant and the second (typical of vector components) being contravariant. The relationship is clearly reflexive in the sense that if we put R = U then R = tl and we can just as well write (10.2.2) as... 

It has been already noted that the rate of a steady-state reaction can be regarded as a vector in the P-dimensional space specified by its components, which are the rates along the basic routes. In terms of linear algebra, the above result means that when the basis of routes is transformed the reaction rate vector along these routes is transformed contravariantly. 

Consider a 3-D domain that can be adequately described by the generalized curvilinear coordinate system (u, v, w) and that its mappings are adequately smooth to allow consistent definitions. Then, any vector F can be decomposed into three components with respect to the contravariant a , a , a or the covariant a , a, a,a, linearly independent basis system as... 

In Seet. 4.2, we need veetor spaee with abasis whieh is formed by A linear independent vectors gp p =, ..., k) which are not generally perpendicular or of unit length [12, 18, 19]. Sueh nonorthogonal basis, we eall a contravariant one. Covariant components of the so called metric tensor are defined by... 

A metric tensor with matrix 9pq is obviously symmetrical and regular (this last assertion is necessary and sufficient for the linear independence of gp in the basis of k orthonormal vectors in this space, we obtain det g , as a product of two determinants first of them having the rows and second one having the columns formed from Cartesian components of gp and gq. Because of the linear independence of these k vectors, every determinant and therefore also det g , is nonzero and conversely). Contravariant components gP of the metric tensor are defined by inversion... 

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