M402-Chapter3

M402-Chapter3 - III-1 Chapter III: Tensors This will be a...

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III-1 Definition : a tensor is an array of covariant and contravariant components , T k ± m np ( functions of x i ) which transforms into a " similar " array of components T ' k ' ± ' m ' n ' p ' ( functions of q i ) as follows : T k ± m np ± A k . k ' A ± ± ' A m . m ' B n n ' B p p ' T ' k ' ± ' m ' n ' p ' Chapter III: Tensors This will be a brief summary of what we have already covered (as it applies to tensors), plus a little about tensors in general. The transformation, A (and its covariant counterpart, B ), can be any linear transformation on the coordinates such as the general coordinate transformations we considered in Chapter II. The most common coordinate transformation in three dimensional Euclidian space is a rotation from one Cartesian system to another. In relativistic problems A is generally a Lorentz transformation from one {x,y,z,t} system to another {x',y',z',t'}. For such transformations the indices (written as greek letters) run from 1 to 4. Recall that: A i j ² ³ x i / ³ q j ...... and the dependence of x i on q j defines the "transformation". Note that a tensor is defined in terms of a transformation , A . Some quantities are tensors (i.e., transform as shown above) only when A is a rotation. Some quantities are tensors under the generalized coordinate transformation. Other quantities are tensors under Lorentz transformations. So when stating that a quantity is a "tensor" one should add "with respect to . ...... (name) transformation". Definition: the rank (contravariant or covariant) of a tensor is equal to the number of components: T k ± mn rp is a mixed tensor with contravariant rank = 4 and covariant rank = 2. Examples 1. Strain tensor w.r.t. rotations between Cartesian systems: S j k = ½ [ ³ δ r j / ³ x k - ³ δ r k / ³ x j ] where δ r j ² ( r 2 - r 1 x^ j and the r i are position vectors.
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III-2 2. Hooke's Law (a "tensor" equation w.r.t. rotations between Cartesian systems): τ jk =C jk ± m S ± m ± stress "tensor" 3. Inertia tensor, I jk , (w.r.t rotations between Cartesian systems): T=½ I jk ω j ω k ; ω ± angular velocity. = kinetic energy of rotation. + I jk ± |- ±±± ρ ( r ) x k x j dV if j ² k | | ½ | ε j ± m | ±±± ρ ( r )[(x ± )²+ (x m )²] dV if j = k . The above are all tensors under orthogonal transformations in three dimensional space, A -1 = A T , and are called affine tensors. Affine ==> cartesian systems. Theorem: Any product of covariant and contravariant vectors which are not operators is tensor under the general linear coordinate transformation: F i G k H ± R m = T i k ± m . The components of T then transform as shown in the box above. SCALAR ± tensor with rank = 0. VECTOR ± tensor with rank (covariant or contravariant) = 1. Addition of tensors: Two tensors of identical rank can be added: M ij + W ij = T ij Note that to each component of M is added the corresponding component of W : M 11 +W 11 = T 11
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III-3 Theorem: the Kronecker Delta, δ i j , is a mixed tensor.
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M402-Chapter3 - III-1 Chapter III: Tensors This will be a...

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