Problem 224 - Calculus Volume 3

Question

Find the divergence of \(\mathbf{F}\). $$ \mathbf{F}(x, y)=(\sin x) \mathbf{i}+(\cos y) \mathbf{j} $$

Step-by-Step Solution

Answer

The divergence of \( \mathbf{F} \) is \( \cos x - \sin y \).

Step 1: Understand the Formula

The divergence of a vector field \( \mathbf{F}(x, y) = P(x, y) \mathbf{i} + Q(x, y) \mathbf{j} \) is given by the formula \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \). In this exercise, identify \( P(x, y) = \sin x \) and \( Q(x, y) = \cos y \).

Step 2: Compute Partial Derivative of P with Respect to x

Calculate \( \frac{\partial P}{\partial x} \) where \( P(x, y) = \sin x \). The derivative of \( \sin x \) with respect to \( x \) is \( \cos x \). Thus, \( \frac{\partial P}{\partial x} = \cos x \).

Step 3: Compute Partial Derivative of Q with Respect to y

Calculate \( \frac{\partial Q}{\partial y} \) where \( Q(x, y) = \cos y \). The derivative of \( \cos y \) with respect to \( y \) is \(-\sin y \). Thus, \( \frac{\partial Q}{\partial y} = -\sin y \).

Step 4: Add the Partial Derivatives

Combine the results from the previous steps to find the divergence of \( \mathbf{F} \): \( abla \cdot \mathbf{F} = \cos x - \sin y \). This is the divergence of the vector field \( \mathbf{F}(x, y) \).

Key Concepts

Vector Field

A vector field is a mathematical construct where each point in space associates a vector. Imagine a wind map where each geographic location has an arrow indicating both the wind's direction and strength. This is similar to a vector field.
In mathematical terms, a vector field in two-dimensional space is often represented as \( \mathbf{F}(x, y) = P(x, y) \mathbf{i} + Q(x, y) \mathbf{j} \). Here, \( P \) and \( Q \) are the components of the vector field, and \( \mathbf{i} \) and \( \mathbf{j} \) are unit vectors in the x and y directions, respectively.

• The vector field in our exercise is \( \mathbf{F}(x, y) = (\sin x) \mathbf{i} + (\cos y) \mathbf{j} \).
• Each point \((x, y)\) has an associated vector determined by \( \sin x \) and \( \cos y \).
• This helps visualize flows like water currents or air flows in physics.

Understanding vector fields is essential in physics and engineering, where they model phenomena such as electromagnetic fields and fluid dynamics.

Partial Derivatives

Partial derivatives are a core concept in calculus, essential for analyzing multi-variable functions.
When dealing with functions of two or more variables, a partial derivative allows us to study how the function changes as one variable changes, keeping others constant. This is akin to examining one dimension at a time.

• In our vector field, we have two components: \( P(x, y) = \sin x \) and \( Q(x, y) = \cos y \).
• The partial derivative of \( P \) with respect to \( x \) is \( \frac{\partial P}{\partial x} = \cos x \).
• For \( Q \), the partial derivative with respect to \( y \) is \( \frac{\partial Q}{\partial y} = -\sin y \).

These derivatives show how the vector field's components change when one of the variables changes, which is crucial for finding the divergence.

Calculus Volume 3

"Calculus Volume 3" likely refers to a textbook focusing on advanced calculus concepts. Topics often include multi-variable calculus, vector analysis, and applications in various fields.
This volume builds on prior knowledge of single-variable calculus and expands the student's ability to model and solve complex situations using higher-dimensional calculus.

• It introduces critical tools like divergence, curl, and gradient, essential for understanding fields.
• The divergence of a vector field is a concept taught in this textbook, showing how much a point acts as a source or sink.
• Understanding divergence helps in studying fluid flow, electromagnetism, and other physical systems.

Working through exercises in "Calculus Volume 3" strengthens students' grasp on these comprehensive topics, deepening their analytical and problem-solving skills.