Problem 20 - The Calculus with Analytic Geometry

Question

\(\int_{C}(\tan y+2 x y \sec z) d x+\left(x \sec ^{2} y+x^{2} \sec z\right) d y+\sec z\left(x^{2} y \tan z-\sec z\right) d z ; A\) is \(\left(2, \frac{1}{4} \pi, 0\right)\) and \(B\) is \((3, \pi, \pi) ;\) Exercise 10

Step-by-Step Solution

Answer

Find the potential function \( \phi \) and compute \( \phi (3, \pi, \pi) - \phi (2, \frac{1}{4} \pi, 0) \).

Step 1: Understanding the Problem

Identify the type of problem. This problem involves evaluating a line integral of a vector field along a curve between points A and B.

Step 2: Identify the given vector field

The given vector field is \[ \textbf{F}(x, y, z) = (\tan y + 2xy \sec z) \hat{i} + (x \sec^2 y + x^2 \sec z) \hat{j} + \sec z (x^2 y \tan z - \sec z) \hat{k} \]

Step 3: Determine if the vector field is conservative

Check if the given vector field is conservative by finding the curl of \[ \textbf{F}\]

Step 4: Calculate the curl of \( \textbf{F} \)

Compute the curl of \( \textbf{F} \) to check if it is zero. The curl is given by: \[ abla \times \textbf{F} = abla \times (\tan y + 2xy \sec z, x \sec^2 y + x^2 \sec z, \sec z (x^2 y \tan z - \sec z)) \]

Step 5: Evaluate the conservativeness

After computing, if \[ abla \times \textbf{F} = 0 \], then \( \textbf{F} \) is conservative.

Step 6: Identification of Potential Function

Since the field is conservative, there exists a potential function \( \phi(x, y, z) \) such that \( \textbf{F} = abla \phi \)

Step 7: Compute Potential Function

To find \( \phi \), we integrate the vector field components: \[ \frac{\partial \phi}{\partial x} = \tan y + 2xy \sec z, \frac{\partial \phi}{\partial y} = x \sec^2 y + x^2 \sec z, \frac{\partial \phi}{\partial z} = \sec z (x^2 y \tan z - \sec z) \]

Step 8: Potential Function Integration

Solve for \( \phi \) by integrating each component step by step, ensuring continuity and consistency.

Step 9: Evaluate the Line Integral

Since \( \int_C \textbf{F} \cdot d\textbf{r} \) for a conservative field is \( \phi(B) - \phi(A) \), compute the potential function \( \phi \) at points A and B.

Step 10: Final Calculation

Finally, substitute points A \( (2, \frac{1}{4}\pi, 0) \) and B \( (3, \pi, \pi) \) to find \( \phi(B) - \phi(A) \)

Key Concepts

Vector Field

In mathematics, a vector field assigns a vector to every point in a space. You can think of it as a field of arrows where each arrow has a direction and a magnitude. For example, the given vector field in this problem is \( \textbf{F}(x, y, z) = (\tan y + 2xy \, \text{sec} \, z) \hat{i} + (x \, \text{sec}^2 \, y + x^2 \, \text{sec} \, z) \hat{j} + \text{sec} \, z(x^2 y \tan z - \text{sec} \, z) \hat{k} \).

Conservative Field

A conservative field is a special type of vector field where the path taken does not affect the work done by the field. In other words, the work done from point A to point B is the same regardless of the path you take. For a field to be conservative, its curl must be zero, which means \( abla \times \textbf{F} = 0 \). If this condition is met, the vector field is said to be conservative.

Potential Function

When a vector field is conservative, there exists a potential function \( \phi(x, y, z) \) such that the field can be expressed as the gradient of \( \phi \), i.e., \( \textbf{F} = abla \phi \). The potential function represents the scalar potential energy at each point in the field. To find \( \phi \), you must integrate the components of the vector field. Each component corresponds to a partial derivative of \( \phi \), so you end up integrating each one to reconstruct \( \phi \).

Curl of a Vector Field

The curl of a vector field measures the tendency of the field to rotate around a point. Mathematically, it is a vector that represents the infinitesimal rotation of a field at a point. For the given vector field \( \textbf{F} \), the curl is computed as: \[ abla \times \textbf{F} = abla \times (\tan y + 2xy \, \text{sec} \, z, x \, \text{sec}^2 \, y + x^2 \, \text{sec} \, z, \text{sec} \, z(x^2 y \tan z - \text{sec} \, z)) \]. If this curl is zero, the field is conservative.