By James G. Simmonds

In this article which progressively develops the instruments for formulating and manipulating the sphere equations of Continuum Mechanics, the math of tensor research is brought in 4, well-separated levels, and the actual interpretation and alertness of vectors and tensors are under pressure all through. This new version comprises extra workouts. furthermore, the writer has appended a bit on Differential Geometry.

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**Example text**

44) are each examples of a coordinate transfonnation of the fonn x = x(u,v,w), y = y(u,v,w), z = z(u,v,w). 46) = X 3 (U I ,U 2 ,U 3 ). Each of these relations is of the same fonn. 47) However, there is no need to continually remind ourselves that we are working in 3-dimensions. Indeed, the beauty of tensor notation is that it reveals those relations that hold in any (finite) number of dimensions. 48) One final compression of notation. Let us agree that in the argument of a function, a sequence of variables such as u I ,u 2 ,u 3 may be replaced by a single symbol, say uj • The only provision is that the symbol for the index (j in this case) be distinct from the symbols for any other indices in the tenn in which Xi appears.

1) is effected analytically as in the following. P. 1. 2 27 The Summation Convention SOLUTION. e. 7), we get 3= 3= VI -VI 6=2v l - + v2 - v3 2v 3 +V 2 +V 3 . Solving these simultaneous linear algebraic equations, we have The Jacobian of a Basis Is Nonzero Unless you are good at perspective, it may be difficult to see from a sketch whether three vectors form a basis. ) Is there a numerical test for a set of vectors to be a basis? Yes, providing we know their Cartesian components. 1 illustrates, the answer hinges on whether a set of n simultaneous linear algebraic equations in n unknowns has a unique solution.

Hint: det AB = (det A) (det B) (If det Q = + 1, Q is called a proper ~ tensor or rotator) (c). , gij and gij) are unchanged. (d). Explain why the columns or rows of Q may be regarded as the Cartesian components of mutually ~ unit vectors. (e). If Q = Q:jgi~ = Qj'g'gi, show that and hence that QkiQ~j = a;. 17. A reflector is an ~ tensor H that reflects vectors across a plane with normal n, as indicated in Fig. 4. (a). Show, geometrically, that H = 1 - 200. 40) Hint: What vector when added to v gives Hv?