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ELEN 3371 Electromagnetics Fall 2008 1 Appendix A: Tensors Instructor: Dr. Gleb V. Tcheslavski Contact: gleb@ee.lamar.edugleb@ee.lamar.edu Office Hours: TR Class web site: http://www.ee.lamar.edu/gleb/ em/Index.htm http://www.ee.lamar.edu/gleb/ em/Index.htm “tensor” by Kevin McCormick

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ELEN 3371 Electromagnetics Fall 2008 2 The Euler’s formulas (A.2.1) (A.2.2) (A.2.3) (A.2.4)

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ELEN 3371 Electromagnetics Fall 2008 3 Phasors (re-visited) Example: Express the loop eqn for a circuit in phasors if v(t) = V 0 cos( t) - The loop eqn. (A.3.1) (A.3.2) Since cos is a reference, we can express the current similarly to (A.3.1): Combining (A.3.1), (A.3.2), and (A.3.3), we arrive at: (A.3.3) (A.3.4)

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ELEN 3371 Electromagnetics Fall 2008 4 Phasors (re-visited 2) Perhaps, we can solve the last equation for I 0 and … Let’s use phasors instead! Recall: then (A.4.1) (A.4.2) Here and are phasors that correspond to the voltage and current respectively. The beauty of this notation is that phasors are free of time dependence! To express the loop equation in phasors, we replace the derivative and the integral in the loop eqn (A.3.4) by j and 1/j respectively. (A.4.3)

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ELEN 3371 Electromagnetics Fall 2008 5 Tensors So far, we have learned that quantities either have a direction (vectors) or they don’t (scalars)… the truth, however, is that a quantity may have MORE THAN ONE direction! A dyadic (also referred to as a dyadic tensor), like a vector, is a quantity that has magnitude and direction but unlike the vector, the dyadic has a dual directionality.

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ELEN 3371 Electromagnetics Fall 2008 6 Tensors (cont) A dyadic in multilinear algebra is formed by juxtaposing pairs of vectors, i.e. placing pairs of vectors side by side. Each component of a dyadic is a dyad. A dyad is the juxtaposition of a pair of basis vectors and a scalar coefficient. As an example: A juxtaposition of A and X is The dyadic tensor is a 90 o rotation operator in 2D. In 3D, the juxtaposition would have 9 components.

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ELEN 3371 Electromagnetics Fall 2008 7 Tensors (cont 2) As we discussed, while a vector has a single direction, a dyad is a dual-directional quantity. A tensor – a multi- directional quantity – is a further generalization. A “dimensionality” of a tensor is called a rank. In fact, a scalar can be viewed as a tensor of rank 0; vector is a tensor of rank 1, a dyadic is a tensor of rank 2, etc. Finally, similarly to scalar fields and vector fields, there are also tensor fields!

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ELEN 3371 Electromagnetics Fall 2008 8 Why bother? General relativity is formulated completely in the language of tensors. For anisotropic dielectrics (those having different physical properties in different directions), such as in crystalline materials, dielectric properties are expressed in terms of tensors. Piezoelectric and magnetostrictive materials used for acoustic transducers are examples of anisotropic materials. Wave propagation in the ionosphere and other plasma media constitute further examples of the use of tensors. Perhaps the most important engineering examples are the stress tensor and strain tensor, which are both 2nd rank tensors, and are related in a general linear material by a fourth rank elasticity tensor.

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