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Differentiation of sums, scalar multiples, inner products, cross products and squared norms of differentiable curves in $R^{3}$

Statement

Let $I \subseteq R$ be an interval with at least two points (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length), let $t_{0} \in I$ , let $x, y : I \to R^{3}$ be differentiable at $t_{0}$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral) and let $λ : I \to R$ be differentiable at $t_{0}$ (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set). Write $x^{'} (t_{0}), y^{'} (t_{0}) \in R^{3}$ for the derivatives, $⟨ \cdot, \cdot ⟩$ for the Euclidean inner product and $∥ \cdot ∥_{2}$ for the Euclidean norm (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ), and $\times$ for the cross product (The cross product in $R^{3}$ ). Then:

$x + y$ and $λ x$ are differentiable at $t_{0}$ , with $(x + y)^{'} (t_{0}) = x^{'} (t_{0}) + y^{'} (t_{0})$ and $(λ x)^{'} (t_{0}) = λ^{'} (t_{0}) x (t_{0}) + λ (t_{0}) x^{'} (t_{0})$ ;
$t \mapsto ⟨ x (t), y (t) ⟩$ is differentiable at $t_{0}$ , with derivative $⟨ x^{'} (t_{0}), y (t_{0}) ⟩ + ⟨ x (t_{0}), y^{'} (t_{0}) ⟩$ ;
$t \mapsto x (t) \times y (t)$ is differentiable at $t_{0}$ , with derivative $x^{'} (t_{0}) \times y (t_{0}) + x (t_{0}) \times y^{'} (t_{0})$ ;
$t \mapsto ∥ x (t) ∥_{2}^{2}$ is differentiable at $t_{0}$ , with derivative $2 ⟨ x^{'} (t_{0}), x (t_{0}) ⟩$ .

Facts & Assumptions

Given: An interval $I \subseteq R$ with at least two points, a point $t_{0} \in I$ , curves $x, y : I \to R^{3}$ and a function $λ : I \to R$ , with $x, y, λ$ differentiable at $t_{0}$ .

[L1]

A function $f : I \to R^{m}$ is differentiable at $t_{0}$ if and only if every component $f_{k}$ is differentiable at $t_{0}$ as a real function on $I$ , and then $f^{'} (t_{0})_{k} = f_{k}^{'} (t_{0})$ ; moreover sums and scalar multiples of differentiable curves are differentiable with $(f + g)^{'} (t_{0}) = f^{'} (t_{0}) + g^{'} (t_{0})$ and $(μ f)^{'} (t_{0}) = μ f^{'} (t_{0})$ for $μ \in R$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

[L2]

Sums, scalar multiples and products of real functions differentiable at $t_{0}$ are differentiable at $t_{0}$ with the sum, scalar-multiple and product rules (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ); the limit-point hypothesis of that theorem is part of the meaning of differentiability at $t_{0}$ (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Limit point, isolated point, adherent point, derived set, and dense subset of $R$ ).

[L3]

The Euclidean inner product and norm are $⟨ u, w ⟩ = \sum_{k < 3} u_{k} w_{k}$ and $∥ u ∥_{2} = \sqrt{⟨ u, u ⟩}$ ; the inner product is bilinear and symmetric (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Laws of finite sums and finite products).

[L4]

For all $u, v \in R^{3}$ the cross product is bilinear-alternating: $(u, v) \mapsto u \times v$ is bilinear, $u \times u = 0$ , and $u \times v = - (v \times u)$ ; in coordinates $(u \times v)_{0} = u_{1} v_{2} - u_{2} v_{1}$ , $(u \times v)_{1} = u_{2} v_{0} - u_{0} v_{2}$ , $(u \times v)_{2} = u_{0} v_{1} - u_{1} v_{0}$ (The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors).

Proof

technique · direct

1.1givenL1L2

By [L1] the components $x_{k}, y_{k} : I \to R$ are differentiable at $t_{0}$ with $x_{k}^{'} (t_{0}) = x^{'} (t_{0})_{k}$ and $y_{k}^{'} (t_{0}) = y^{'} (t_{0})_{k}$ , and by [L2] each of the functions $x_{k} + y_{k}$ , $λ x_{k}$ , $x_{k} y_{k}$ is differentiable at $t_{0}$ with the sum, scalar-multiple and product rules; this proves clause 1 componentwise, since $(x + y)_{k} = x_{k} + y_{k}$ and $(λ x)_{k} = λ x_{k}$ .

2.1step 1.1L2L3

The scalar function $t \mapsto ⟨ x (t), y (t) ⟩ = \sum_{k < 3} x_{k} (t) y_{k} (t)$ is a finite sum of products of functions differentiable at $t_{0}$ , hence by [L2] differentiable at $t_{0}$ with derivative $\sum_{k < 3} (x_{k}^{'} (t_{0}) y_{k} (t_{0}) + x_{k} (t_{0}) y_{k}^{'} (t_{0}))$ ; by [L3] this is $⟨ x^{'} (t_{0}), y (t_{0}) ⟩ + ⟨ x (t_{0}), y^{'} (t_{0}) ⟩$ , which is clause 2.

2.2step 1.1L1L2L4

Each component of $t \mapsto x (t) \times y (t)$ is, by [L4], one of the differences $x_{1} y_{2} - x_{2} y_{1}$ , $x_{2} y_{0} - x_{0} y_{2}$ , $x_{0} y_{1} - x_{1} y_{0}$ of products of functions differentiable at $t_{0}$ , so [L2] makes it differentiable at $t_{0}$ with derivative $(x_{1}^{'} y_{2} + x_{1} y_{2}^{'}) - (x_{2}^{'} y_{1} + x_{2} y_{1}^{'})$ , and by [L4] and bilinearity this is the corresponding component of $x^{'} (t_{0}) \times y (t_{0}) + x (t_{0}) \times y^{'} (t_{0})$ ; reassembling the three components with [L1] gives clause 3.

3.1step 2.1L3algebra∎

Applying clause 2 at $y : = x$ gives differentiability of $t \mapsto ⟨ x (t), x (t) ⟩ = ∥ x (t) ∥_{2}^{2}$ with derivative $⟨ x^{'} (t_{0}), x (t_{0}) ⟩ + ⟨ x (t_{0}), x^{'} (t_{0}) ⟩ = 2 ⟨ x^{'} (t_{0}), x (t_{0}) ⟩$ , the two terms being equal by symmetry of the inner product [L3]; this is clause 4.

Differentiation of sums, scalar multiples, inner products, cross products and squared norms of differentiable curves in $R^{3}$

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Differentiation of sums, scalar multiples, inner products, cross products and squared norms of differentiable curves in R3R3

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Differentiation of sums, scalar multiples, inner products, cross products and squared norms of differentiable curves in $R^{3}$