Problem 38 - Advanced Engineering Mathematics

Question

Suppose $$ \begin{gathered} D_{\mathrm{u}} f(a, b)=7, \quad D_{\mathrm{v}} f(a, b)=3 \\ \mathbf{u}=\frac{5}{13} \mathbf{i}-\frac{12}{13} \mathbf{j}, \quad \mathbf{v}=\frac{5}{13} \mathbf{i}+\frac{12}{13} \mathbf{j} \end{gathered} $$ Find \(\nabla f(a, b)\).

Step-by-Step Solution

Answer

The gradient \( \nabla f(a, b) = \langle 13, -\frac{13}{6} \rangle \).

Step 1: Understand the Gradient Definition

The gradient of a function \( f \) with respect to variables \( x \) and \( y \) is represented as \( abla f(x, y) = \langle f_x, f_y \rangle \), where \( f_x \) and \( f_y \) are the partial derivatives. Our task is to find \( abla f(a, b) = \langle f_x(a, b), f_y(a, b) \rangle \).

Step 2: Use the Directional Derivative Formula

The directional derivative of \( f \) in the direction of \( \mathbf{u} \) is given by the dot product of the gradient with \( \mathbf{u} \). It can be expressed as:\[D_{\mathbf{u}} f(a, b) = abla f(a, b) \cdot \mathbf{u} = 7.\]Similarly, for direction \( \mathbf{v} \), it is expressed as:\[D_{\mathbf{v}} f(a, b) = abla f(a, b) \cdot \mathbf{v} = 3.\]

Step 3: Write the Gradient Dot Product Equations

Since \( \mathbf{u} = \frac{5}{13} \mathbf{i}-\frac{12}{13} \mathbf{j} \) and \( \mathbf{v} = \frac{5}{13} \mathbf{i}+\frac{12}{13} \mathbf{j} \), the dot products can be written as:\[abla f(a, b) \cdot \left( \frac{5}{13}, -\frac{12}{13} \right) = \frac{5}{13}f_x(a, b) - \frac{12}{13}f_y(a, b) = 7,\]\[abla f(a, b) \cdot \left( \frac{5}{13}, \frac{12}{13} \right) = \frac{5}{13}f_x(a, b) + \frac{12}{13}f_y(a, b) = 3.\]

Step 4: Solve the System of Equations for \(f_x\) and \(f_y\)

We have the following linear system of equations:1. \( \frac{5}{13}f_x - \frac{12}{13}f_y = 7 \)2. \( \frac{5}{13}f_x + \frac{12}{13}f_y = 3 \)To solve, add the two equations to eliminate \( f_y \):\[\left(\frac{5}{13} + \frac{5}{13}\right)f_x = 10,\]which simplifies to \( 2 \cdot \frac{5}{13} f_x = 10 \) and further simplifies to:\[f_x = 13.\]Substitute \( f_x = 13 \) into one of the original equations to find \( f_y \):\[\frac{5}{13} \cdot 13 - \frac{12}{13} f_y = 7 \5 - \frac{12}{13} f_y = 7 \-\frac{12}{13} f_y = 2 \f_y = -\frac{13}{6}.\]

Step 5: State the Gradient

Thus, the gradient \( abla f(a, b) = \langle f_x(a, b), f_y(a, b) \rangle \) is:\[abla f(a, b) = \langle 13, -\frac{13}{6} \rangle.\]

Key Concepts

Partial Derivatives

Partial derivatives are an essential concept in calculus, especially when dealing with functions of multiple variables. Simply put, a partial derivative measures how a function changes as one of its input variables changes, while keeping the other variables constant. Imagine a function like a surface in three-dimensional space. If you move along one axis (say the x-axis) while keeping the y-value fixed, the slope you encounter is the partial derivative with respect to x, denoted as \( f_x(x, y) \).
For functions \( f(x, y) \), we often compute \( f_x \) and \( f_y \), which give us the rates of change in the x and y directions, respectively. These derivatives help in understanding the behavior of the function locally, much like a zoomed-in look on a map.
Partial derivatives are foundational in forming the gradient of a function, which provides comprehensive information about the function's slope in any direction at a specific point. They are also widely used in optimization problems and are vital in fields such as economics, physics, and engineering.

Directional Derivatives

Directional derivatives take the concept of partial derivatives a step further by allowing us to explore the rate of change of a function in any given direction, not just along the coordinate axes. This is akin to moving across any direction on a hill and asking how steep the hill is in that particular direction.
To calculate a directional derivative \( D_{\mathbf{u}} f(a, b) \), we utilize both the gradient of a function and a unit direction vector \( \mathbf{u} \). The formula is:

• \( D_{\mathbf{u}} f(a, b) = abla f(a, b) \cdot \mathbf{u} \)

This is the dot product of the gradient vector and the direction vector \( \mathbf{u} \), providing a scalar that represents the rate of change of the function in that direction. A key aspect is ensuring that \( \mathbf{u} \) is a unit vector, ensuring coherence in the calculations.
Directional derivatives offer a deeper understanding of how functions behave, helping in fields such as machine learning and physics where understanding component behaviors in particular directions is crucial.

Vector Calculus

Vector calculus serves as a robust framework for analyzing vector fields and scalar functions in multi-dimensional spaces. It seamlessly integrates concepts from differential calculus, like partial and directional derivatives, with linear algebra.
In vector calculus, vectors represent quantities with both magnitude and direction, essential for modeling real-world phenomena such as fluid flow, electromagnetism, and more. The gradient \( abla f \) itself is a vector, encoding the direction of the greatest rate of increase for a scalar field \( f \).
Key operations in vector calculus include:

• Gradient: Indicates the direction and rate of change.
• Divergence: Measures a vector field's tendency to converge or diverge from a point.
• Curl: Describes the rotation of a vector field.

Through vector calculus, we can solve complex problems in physics like Maxwell's equations and fluid dynamics, helping to predict and understand the behavior of multi-dimensional systems.