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Chapter 3: Problem 7

Define the derivative \(\mathrm{w}\) (f) of a differentiable function \(\mathrm{f}: \mathrm{U} \subset \mathrm{S} \rightarrow \mathrm{R}\) relative to a vector field \(\mathrm{w}\) in \(\mathrm{U}\) by $$ \mathrm{w}(\mathrm{f})(\mathrm{q})=\left.\frac{\mathrm{d}}{d t}(\mathrm{f} \circ \alpha)\right|_{\mathrm{t}=0}, \quad \mathrm{q} \in \mathrm{U} $$ where \(\alpha: I \rightarrow S\) is a curve such that \(\alpha(0)=q, \alpha^{\prime}(0)=w(q)\). Prove that a. \(w\) is differentiable in \(U\) if and only if \(w(f)\) is differentiable for all differentiable \(f\) in \(U\). b. Let \(\lambda\) and \(\mu\) be real numbers and \(g: U \subset S \rightarrow R\) be a differentiable function on \(U\); then $$ \begin{aligned} &w(\lambda f+\mu f)=\lambda w(f)+\mu w(f) \\ &w(f g)=w(f) g+f w(g) \end{aligned} $$

Short Answer

Expert verified
Part (a) shows differentiability of \(w\) iff \(w(f)\) is differentiable for all \(f\). Part (b) proves \(w\) is linear and satisfies the product rule.

Step by step solution

01

Understand the definition

The derivative of a differentiable function \(f\) relative to a vector field \(w\) is given by \[ w(f)(q) = \left.\frac{d}{dt}(f \circ \alpha)\right|_{t=0} \] where \(\alpha\) is a curve such that \(\alpha(0) = q\) and \(\alpha'(0) = w(q)\). This means that we are evaluating the change in \(f\) along the direction of the vector field \(w\).
02

Prove part (a)

To prove that \(w\) is differentiable in \(U\) if and only if \(w(f)\) is differentiable for all differentiable functions \(f\) in \(U\),.First, assume that \(w\) is differentiable. Then, for any differentiable function \(f\), \(w(f) = df(w)\), where \(df\) denotes the differential of \(f\). Since \(df\) is differentiable and \(w\) is differentiable, their composition \(w(f)\) is also differentiable.Conversely, assume that \(w(f)\) is differentiable for all differentiable functions \(f\) in \(U\). Consider the coordinate functions \(x_i: U \rightarrow \mathbb{R}\), where each \(x_i\) is differentiable. Then, for each \(i\), \(w(x_i)\) is differentiable. Since \(w = \sum_i w(x_i) \frac{\partial}{\partial x_i}\) and each \({\partial}/{\partial x_i}\) is differentiable, it follows that \(w\) must be differentiable.
03

Prove part (b) for linearity

Given \(w(\lambda f + \mu g) = \lambda w(f) + \mu w(g)\), we need to show linearity. Using the definition, we have: \[ w(\lambda f + \mu g)(q) = \left.\frac{d}{dt}[(\lambda f + \mu g) \circ \alpha]\right|_{t=0} \]This simplifies to:\[ \lambda \left.\frac{d}{dt}(f \circ \alpha)\right|_{t=0} + \mu \left.\frac{d}{dt}(g \circ \alpha)\right|_{t=0} \] which equals:\[ \lambda w(f)(q) + \mu w(g)(q) \]
04

Prove part (b) for the product rule

Given \(w(fg) = w(f)g + fw(g)\). Using the definition, we need to show the product rule:\[ w(fg)(q) = \left.\frac{d}{dt}[(f \circ \alpha)(g \circ \alpha)]\right|_{t=0} \]Applying the product rule for derivatives:\[ = \left.\frac{d}{dt}(f \circ \alpha)\cdot g(\alpha(t)) + (f \circ \alpha)\cdot \frac{d}{dt}(g \circ \alpha)\right|_{t=0} \]Evaluating at \(t=0\), we get:\[ = w(f)(q) g(q) + f(q) w(g)(q) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differentiable Functions
In calculus, differentiable functions are those that have derivatives at all points within their domain. This means the function must be smooth and continuous, with no sharp corners or breaks. To be mathematically precise, a function \( f \) is differentiable at a point \( q \) if the limit \[ \frac{f(q + h) - f(q)}{h} \rightarrow L \] as \( h \) approaches zero exists and is finite. The value \( L \) becomes the derivative \( f'(q) \).
Differentiable functions are the building blocks of many concepts in higher mathematics, including vector fields and the product rule, which we'll discuss next.
Vector Fields
A vector field assigns a vector to every point in a subset of space \( U \). Imagine a fluid flowing through a region: at each point, the velocity of the fluid can be described by a vector. In mathematics, a vector field \( w \) is often represented as \[ w = w_1(x, y, z) \frac{\text{鈭倉}{\text{鈭倉x} + w_2(x, y, z) \frac{\text{鈭倉}{\text{鈭倉y} + w_3(x, y, z) \frac{\text{鈭倉}{\text{鈭倉z} \] where \( w_1 \), \( w_2 \), and \( w_3 \) are functions of the spatial coordinates \(x\), \(y\), and \(z\).
In the context of differentiable functions, we often want to understand how the function changes along the direction specified by the vector field. This leads us to the derivative of the function relative to the vector field, which is denoted \( w(f) \). It's a way to capture how \( f \) changes when you move in the direction of \( w \).
Product Rule
The product rule is a fundamental property in calculus used to find the derivative of the product of two functions. If you have two differentiable functions \( f \) and \( g \), the product rule states: \[ (fg)' = f'g + fg' \] This means, to find the derivative of the product of \( f \) and \( g \), you differentiate \( f \) and multiply it by \( g \), then differentiate \( g \) and multiply it by \( f \), and finally add the two results together.
When extended to vector fields, the product rule applies similarly. For functions \( f \) and \( g \) and a vector field \( w \), \[ w(fg) = w(f)g + fw(g) \] as shown in the exercise. This rule helps to simplify and solve more complicated differentiation problems involving products of functions.
Linearity in Calculus
Linearity is a key concept in calculus, especially when dealing with operations on functions and vector fields. A function or operator \( w \) is linear if it satisfies two main properties:
  • Additivity: \[ w(f + g) = w(f) + w(g) \]
  • Homogeneity: \[ w(af) = a w(f) \] for any scalar \( a \).

These properties make it much easier to work with differentiable functions. For instance, when dealing with vector fields, linearity allows us to distribute the differentiation operator \( w \) over sums and scalar multiples, as shown in the exercise with \( w( \lambda f + \mu g) = \lambda w(f) + \mu w(g) \). This linear behavior is a cornerstone of analysis and greatly simplifies many mathematical operations in calculus and beyond.

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Most popular questions from this chapter

Determine the asymptotic curves and the lines of curvature of \(z=x y\).

(Theorem of Beltrami-Enneper.) Prove that the absolute value of the torsion \(\tau\) at a point of an asymptotic curve, whose curvature is nowhere zero, is given by $$ |\tau|=\sqrt{-K}, $$ where \(K\) is the Gaussian curvature of the surface at the given point.

Determine the asymptotic curves and the lines of curvature of the helicoid \(x=v \cos u, y=v \sin u, z=c u\), and show that its mean curvature is zero.

Example 4 can be generalized as follows. A one-parameter differentiable family of planes \(\\{\alpha(t), N(t)\\}\) is a correspondence which assigns to each \(t \in I\) a point \(\alpha(t) \in R^{3}\) and a unit vector \(N(t) \in R^{3}\) in such a way that both \(\alpha\), and \(N\) are differentiable maps. A family \(\\{\alpha(t), N(t)\\}, t \in I\), is said to be a family of tangent planes if \(\alpha^{\prime}(t) \neq 0, N^{\prime}(t) \neq 0\), and \(\left\langle\alpha^{\prime}(t), N(t)\right\rangle=0\) for all \(t \in I\). a. Give a proof that a differentiable one-parameter family of tangent planes \(\\{\alpha(t), N(t)\\}\) determines a differentiable one-parameter family of lines \(\left\\{\alpha(t),\left(N \wedge N^{\prime}\right) /\left|N^{\prime}\right|\right\\}\) which generates a developable surface $$ \mathbf{x}(t, v)=\alpha(t)+v \frac{N \wedge N^{\prime}}{\left|N^{\prime}\right|} . $$ The surface (*) is called the envelope of the family \(\\{\alpha(t), N(t)\\}\). b. Prove that if \(\alpha^{\prime}(t) \wedge\left(N(t) \wedge N^{\prime}(t)\right) \neq 0\) for all \(t \in I\), then the envelope (*) is regular in a neighborhood of \(v=0\), and the unit normal vector of \(\mathbf{x}\) at \((t, 0)\) is \(N(t)\). c. Let \(\alpha=\alpha(s)\) be a curve in \(R^{3}\) parametrized by arc length. Assume that the curvature \(k(s)\) and the torsion \(\tau(s)\) of \(\alpha\) are nowhere zero. Prove that the family of osculating planes \(\\{\alpha(s), b\\{s)\\}\) is a oneparameter differentiable family of tangent planes and that the envelope of this family is the tangent surface to \(\alpha(s)\) (cf. Example 5, Sec. 2-3).

Let \(\mathbf{x}=\mathbf{x}(u, v)\) be a regular parametrized surface. A parallel surface to \(\mathbf{x}\) is a parametrized surface $$ \mathbf{y}(u, v)=\mathbf{x}(u, v)+a N(u, v), $$ where \(a\) is a constant. a. Prove that \(y_{\alpha} \wedge y_{v}=\left(1-2 H a+K a^{2}\right)\left(\mathbf{x}_{u} \wedge \mathbf{x}_{v}\right)\), where \(K\) and \(H\) are the Gaussian and mean curvatures of \(\mathbf{x}\), respectively. b. Prove that at the regular points, the Gaussian curvature of \(\mathbf{y}\) is $$ \frac{K}{1-2 H a+K a^{2}} $$ and the mean curvature of \(\mathbf{y}\) is $$ \frac{H-K a}{1-2 H a+K a^{2}} $$ c. Let a surface \(\mathbf{x}\) have constant mean curvature equal to \(c \neq 0\) and consider the parallel surface to \(\mathbf{x}\) at a distance \(1 / 2 c\). Prove that this parallel surface has constant Gaussian curvature equal to \(4 c^{2}\).
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