Problem 180 - Advanced Calculus

Question

Let \(\mathrm{f} \in \mathrm{C}^{1}(\mathrm{E})\), the set of all continuously differentiable functions with domain \(E, E\) open in \(R^{n}\), and let \(D_{v} f(p)\) denote the directional derivative in the direction \(\mathrm{v}\) at \(\mathrm{p} \in \mathrm{E}\). What is \(\max _{|| v \mid=1}\left|D_{v} f(p)\right|\) and for which v's of norm 1 is the maximum achieved?

Step-by-Step Solution

Answer

The maximum value of the directional derivative for a continuously differentiable function \(f\) is given by the magnitude of the gradient vector at point \(p\): \(\max _{|| v ||=1} \, |D_{v} f(p)| = || \nabla f(p) ||\) This maximum is achieved for the direction vector \(v\) with norm 1, which is: \(v = \frac{\nabla f(p)}{|| \nabla f(p) ||}\)

Step 1: Find the gradient vector of f at point p

To find the maximum directional derivative, we first need to find the gradient vector, ∇f(p), of the function f at a given point p. The gradient vector is a collection of partial derivatives, which can be written as: \(\nabla f(p) = \begin{bmatrix}\frac{\partial{f}}{\partial{x_1}}(p)\\\frac{\partial{f}}{\partial{x_2}}(p)\\\vdots\\\frac{\partial{f}}{\partial{x_n}}(p)\end{bmatrix}\)

Step 2: Find the directional derivative in terms of the gradient vector

Now that we have the gradient vector, we can find the directional derivative of f in the direction of v: \(D_v f(p) = \nabla f(p) \cdot v = || \nabla f(p) || \, ||v|| \, cos(\theta)\) where θ is the angle between ∇f(p) and v, and ||∇f(p)|| and ||v|| are the magnitudes of the respective vectors. Since v has a norm of 1, ||v|| = 1.

Step 3: Find the maximum directional derivative

The maximum directional derivative is achieved when the direction vector v is parallel to the gradient vector ∇f(p), which means the angle between them is 0, and therefore, cos(θ) = 1. So, the maximum directional derivative is: \(\max _{|| v ||=1} \, |D_{v} f(p)| = || \nabla f(p) ||\)

Step 4: Determine the direction vector v for which the maximum is achieved

To find the direction vector v with norm 1 for which the maximum directional derivative is achieved, we need to normalize the gradient vector ∇f(p): \(v = \frac{\nabla f(p)}{|| \nabla f(p) ||}\) So, the maximum value of the directional derivative is given by the magnitude of the gradient vector: \(\max _{|| v ||=1} \, |D_{v} f(p)| = || \nabla f(p) ||\) and the maximum is achieved for the direction vector v with norm 1, which is: \(v = \frac{\nabla f(p)}{|| \nabla f(p) ||}\)

Key Concepts

Gradient Vector

The gradient vector is an essential concept when we talk about directional derivatives and continuously differentiable functions. In simple terms, the gradient vector, denoted as \( abla f(p) \), represents how a function changes at a point \( p \) across multiple directions. Think of it as a vector that points in the direction of the steepest ascent of the function from that point.

• It is composed of the partial derivatives of the function with respect to each variable, indicating how the function changes along those axes.
• This vector provides crucial information about the shape and orientation of the function's graph around the point \( p \).
• The magnitude of the gradient vector, \( || abla f(p) || \), tells us how steep this ascent is.

Understanding the gradient vector is key to solving problems involving maximum directional derivatives because it helps determine the direction in which the function increases most rapidly. The magnitude of the gradient vector serves as the maximum value of the directional derivative when the direction is perfectly aligned with the gradient.

Maximum Norm

The maximum norm helps us determine the bounds of directional derivatives for functions. In optimization and calculus, the norm of a vector is used to describe its size or length. The maximum norm specifically asks us to find the maximum possible value of the directional derivative, which occurs in the direction of the gradient vector.

• When computing \( \max_{||v||=1} |D_v f(p)| \), we're looking for the value of the directional derivative that has the greatest magnitude when the direction vector \( v \) is a unit vector.
• The condition \(||v|| = 1\) ensures that we focus on the unit direction vectors because vectors of different lengths could falsely inflate the derivative's value.

In the context of directional derivatives, the maximum norm simplifies to the magnitude of the gradient vector \( || abla f(p) || \). This is because the largest derivative is attained when the direction of \( v \) aligns with the gradient vector, making the angle between them zero and \( \cos(\theta) = 1 \). This means the direction of the steepest slope gives the maximum change rate per unit distance.

Continuously Differentiable Functions

Continuously differentiable functions are the backbone of smooth calculus operations, and understanding them is crucial when working with gradient vectors and directional derivatives. A function is continuously differentiable if it has derivatives that do not jump or have inconsistencies, and can be smoothly traced across its domain. This is denoted as \( C^1(E) \), emphasizing that both the function and its first derivatives are continuous over an open set \( E \).

The major advantages of working with continuously differentiable functions include:

• Smoothness: They allow a seamless application of calculus operations, such as finding limits, gradients, and directional derivatives.
• Predictable behavior: Since derivatives change smoothly, the function doesn't experience sudden spikes or dips which are usually hard to manage analytically.
• Analysis conveniences: They offer strong guarantees about how functions behave locally around a point, which is essential for many analytical techniques in optimization and approximation.

Overall, when considering directional derivatives, the condition of being continuously differentiable ensures that not only can we compute these derivatives, but we can also consider how they vary continuously with changes in direction or position.