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Topics from Tensoral Calculus∗

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Topics from Tensoral Calculus∗ Topics from Tensoral Calculus∗ Jay R. Walton Fall 2013 1 Preliminaries These notes are concerned with topics from tensoral calculus, i.e. generalizations of calculus to functions defined between two tensor spaces. To make the discussion more concrete, the tensor spaces are defined over ordinary Euclidean space, RN , with its usual inner product structure. Thus, the tensor spaces inherit a natural inner product, the tensor dot-product, from the underlying vector space, RN . 2 The Derivative of Tensor Valued Functions Let F : D ⊂ T r −! T s (1) be a function with domain D a subset of T r taking values in the tensor space T s, where D is assumed to be an open subset of T r.1 Continuity. The function F is said to be Continuous at A 2 D provided for every > 0 there exists a δ > 0 such F (B) 2 B(F (A); ) whenever B 2 B(A; δ), i.e. F maps the ball of radius δ centered at A into the ball of radius centered at F (A). The function F is said to be continuous on all of D provided it is continuous at each A 2 D. There are two useful alternative characterizations of continuity. The first is that F is continuous on D provided it maps convergent sequences onto convergent sequences. That is, if An 2 D is a sequence converging to A 2 D (limn!1 An = A), then limn!1 F (An) = F (A). The second alternative characterization is that the inverse function F −1 maps open subsets of T s to open subsets of D. Derivative. The function F is said to be Differentiable at A 2 D provided there exists a tensor L 2 T r+s such that F (A + H) = F (A) + L[H] + o(H) as jHj ! 0 (2) ∗Copyright c 2011 by J. R. Walton. All rights reserved. 1A subset D ⊂ T r is said to be open provided for every element A 2 D there exists an open ball centered at A that is wholly contained in D. 1 where jHj denotes the norm of the tensor H 2 T r. If such a tensor L exists satisfying (??), it is called the Derivative of F at A and denoted DF (A). Thus, (??) can be rewritten F (A + H) = F (A) + DF (A)[H] + o(H): (3) Recall that o(jHj) is the Landau \little oh" symbol which is used to denote a function depending upon H that tends to zero faster that jHj, i.e. o(H) lim = 0: jHj!0 jHj If the derivative DF (A) exists at each point in D, then it defines a function DF (·): D ⊂ T r −! T r+s: Moreover, if the function DF (·) is differentiable at A 2 T r, then its derivative is a tensor in T 2r+s, denoted by D2F (A), called the second derivative of F at A 2 D. Continuing in this manner, derivatives of F of all orders can be defined. Example. Let 1 ∼ N 0 ∼ φ(·): T = R −! T = R: (4) Thus, φ(·) is a real-valued function of N-real variables and its graph is a surface in RN+1. In the definition of the derivative (??), it is more customary in this context to let H = hu where u is a unit vector in RN . Defining equation (??) then becomes φ(a + hu) = φ(a) + Dφ(a)[hu] + o(hu): (5) From the linearity of the tensor Dφ(a)[hu], one concludes that φ(a + hu) − φ(a) Dφ(a)[u] = lim ; (6) h!0 h which is the familiar directional derivative of φ(·) at the point a in the direction u. Thus, being differentiable at a point implies the existence of directional derivatives, and hence partial derivatives, in all directions. However, the converse is not true. That is, there exist functions with directional derivatives existing at a point in all possible directions but which are not differentiable at the point. For such an example, consider the function φ(·): R2 −! R given by ( 3 3 x −y when (x; y) 6= (0; 0); φ(x; y) = x2+y2 (7) 0 when (x; y) = (0; 0): One then shows easily that if u = (cos(θ); sin(θ))T , the directional derivative of φ(·) at the origin (0,0) in the direction u equals cos(θ)3 − sin(θ)3. However, φ(·) is not differentiable at the origin in the sense of (??). (Why?) Consider further the function φ(·) in (??). If φ(·) is differentiable in the sense of (??), then its derivative Dφ(a)[·] 2 T 1 is a linear transformation from RN to R, and as such is representable by dot-product with a vector in RN . Specifically, there exists a unique vector, denoted by rφ(a) 2 RN such that Dφ(a)[u] = rφ(a) · u for all u 2 RN : (8) 2 The vector rφ(a) is called the Gradient of φ(·) at a. The component forms for the derivative Dφ and the gradient rφ are easily constructed. N In particular, let B = fe1;:::; eN g be the natural orthonormal basis for R . Then the 1×N matrix representation for Dφ and the N-tuple vector representation for the gradient rφ are given by 2 3 @x1 φ(a) [Dφ(a)] = [@ φ(a);:::;@ φ(a)] and [rφ(a)] = 6 . 7 (9) B x1 xN B 4 . 5 @xN φ(a) where @xi φ(a) denotes the partial derivative of φ with respect to xi at the point a φ(a + hei) − φ(a) @x φ(a) = lim : i h!0 h Example. Another important example is provided by functions F (·): T 0 −! T s, i.e. s-ordered tensor valued functions of a single scalar variable. Since in continuum mechanics the scalar independent variable is usually time, that variable is given the special symbol t and the derivative of such functions is represented by DF (t)[τ] = F_ (t)τ 2 T s (10) where F (t + h) − F (t) F_ (t) = lim : h!0 h In component form, if the tensor valued function F (·) has the component representation s [Fi1;:::;is ] with respect to the natural basis for T , then the component representation for the tensor F_ (t) is _ _ [F (t)] = [Fi1;:::;is ]: (11) 1 ∼ N 1 Example. A Vector Field is a function a(·): D ⊂ T = R −! T . Its derivative defines a second order tensor Da(x)[·] 2 T 2. Its component form, with respect to the natural basis on RN for example, is [Da(x)] = [@xj ai(x)] i; j = 1;:::;N (12) where [ai(x)]; i = 1;:::;N gives the component representation of a(x). The right hand side of (??) is the familiar Jacobian matrix. Product Rule. Various types of \products" of tensor functions occur naturally in tensor calculus. Rather than proving a separate product rule formula for every product that arises, it is much more expedient and much cleaner to prove one product rule formula for a general, abstract notion of product. To that end, the appropriate general notion of product is provided by a general bi-linear form. More specifically, suppose that F (·): D ⊂ T r −! T p and G(·): D ⊂ T r −! T q are two differentiable functions with the same domain set D in T r but different range spaces. Letπ ^(·; ·): T p × T q −! T s denote a bi-linear function (i.e. π^(·; ·) is linear in each of its variables separately) with values in T s. One then defines the product function E(·): D ⊂ T r −! T s by E(A) :=π ^(F (A);G(A)) for A 2 D: 3 Since F and G are assumed to be differentiable at A 2 D, it is not difficult to show that E is also differentiable at A with DE(A)[H] =π ^(DF (A)[H];G(A)) +π ^(F (A); DG(A)[H]) for all H 2 T r. (13) Notice that (??) has the familiar form (fg)0 = f 0g + fg0 from single variable calculus. 0 ∼ r 0 s Example. Let A(·): T = R −! T and B(·): T −! T be differentiable tensor valued functions of the single scalar variable t. Then their tensor product E(t) := A(t) ⊗ B(t) is differentiable with E_ (t) = A_(t) ⊗ B(t) + A(t) ⊗ B_ (t): Chain Rule The familiar chain rule from single variable calculus has a straight forward generalization to the tensor setting. Specifically, suppose F (·): D ⊂ T r −! T q is differ- entiable at A 2 D and G(·): G ⊂ T q −! T s (with G being an open set on which G(·) is defined) is differentiable at F (A) 2 G \ F (D), then the composite function E(·) := G ◦ F (·) is also differentiable at A 2 D with DE(A)[H] = DG(F (A))[DF (A)[H]] for all H 2 T r: (14) The right hand side of (??) is the composition of the tensor DG(F (A))[·] 2 T q+s with the tensor DF (A)[·] 2 T r+q producing a the tensor DG(F (A)) ◦ DF (A)[·] 2 T r+s. This generalizes the familiar chain rule formula g(f(x))0 = g0(f(x))f 0(x) from single variable calculus. Example. An important application of the chain rule is to composite functions of the form E(t) = G ◦ F (t) = G(F (t)), i.e. functions for which the inner function is a function of the single scalar variable t. The chain rule then yields the result E_ (t) = DG(F (t))[F_ (t)]: For example, let A(t) be a differentiable function taking values in T 2, i.e.
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