Problem 26 - Thomas Calculus

Question

Independence of path Show that the values of the integrals do not depend on the path taken from \(A\) to \(B\) . \(\int_{A}^{B} \frac{x d x+y d y+z d z}{\sqrt{x^{2}+y^{2}+z^{2}}}\)

Step-by-Step Solution

Answer

The integral is path independent because the vector field is conservative.

Step 1: Understand the Problem

We need to demonstrate that the line integral is path independent. The integral is of the form \( \int_{A}^{B} \mathbf{F} \cdot d\mathbf{r} \), where \( \mathbf{F} = \left(\frac{x}{\sqrt{x^2+y^2+z^2}}, \frac{y}{\sqrt{x^2+y^2+z^2}}, \frac{z}{\sqrt{x^2+y^2+z^2}} \right) \).

Step 2: Check for Conservative Field

A vector field \( \mathbf{F} \) is conservative if its curl is zero. Compute \( abla \times \mathbf{F} \) to verify if it is zero. Use the components \( P = \frac{x}{\sqrt{x^2+y^2+z^2}} \), \( Q = \frac{y}{\sqrt{x^2+y^2+z^2}} \), \( R = \frac{z}{\sqrt{x^2+y^2+z^2}} \). The formulas for the curl components are:\[\left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\]

Step 3: Calculate Curl of F

Calculate each term of the curl. For a conservative field, each component should equal zero.- The computation involves partial derivatives, which are a bit lengthy but can be verified as follows: - \( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = 0 \) - \( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} = 0 \) - \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0 \)This indicates \( abla \times \mathbf{F} = (0,0,0) \).

Step 4: Conclude Independence of Path

Since \( abla \times \mathbf{F} = \mathbf{0} \), \( \mathbf{F} \) is a conservative vector field. Therefore, the line integral \( \int_{A}^{B} \mathbf{F} \cdot d\mathbf{r} \) does not depend on the path taken from \( A \) to \( B \).

Step 5: State the Result

The integral is determined solely by the endpoints, confirming that the integral's value is path independent.

Key Concepts

Conservative Vector Field

In the world of vector calculus, a **conservative vector field** is an important concept. It applies to scenarios where the vector field determines a potential energy's gradient. One essential property of a conservative vector field is path independence. This means that the total work done in moving an object from one point to another in such a field depends only on the starting and ending points, not on the actual path taken between them.

• A vector field is conservative if it can be expressed as the gradient of a scalar field.
• Another key characteristic is that the line integral around any closed loop in a conservative field is zero.
• An essential mathematical tool to determine if a vector field is conservative is the curl operation, which we will explore further.

Understanding conservative vector fields is crucial for many areas like physics, particularly in analyzing gravitational and electric fields.

Line Integrals

Line integrals are a powerful tool in calculus, allowing us to integrate functions along a curve. They are particularly useful for finding the work done by a force field, such as when it's conservative.
In simple terms, when you compute a line integral, you're summing up a field's contribution along a specific path. The line integral \[\int_{A}^{B} \mathbf{F} \cdot d\mathbf{r}\]represents the accumulated sum of the vector field \( \mathbf{F} \) along the curve from point \( A \) to point \( B \).

• If the vector field is conservative, the value of the line integral depends only on the start and end points.
• This is directly related to path independence in a conservative field.
• To make calculations easier, paths can often be simplified when evaluating these integrals in conservative fields.

Therefore, line integrals form the basis for understanding work done in traversing between two points in a field.

Curl of a Vector Field

The **curl** of a vector field is a vector operator that describes the rotation of a vector field in three-dimensional space. Physically, it measures the tendency of the field to induce rotation around a point, much like how a whirlpool swirls water.
Calculating the curl involves taking the cross product of the del operator (often denoted as \( abla \)) with the vector field \( \mathbf{F} \).

• The result of the curl operation is itself a vector, which indicates the axis of rotation and the magnitude of the rotation.
• A key property is: A vector field is conservative if its curl is zero everywhere in the domain where the field is defined.
• This simplification arises because a zero curl implies no intrinsic rotation, aligning with the notion of path independence in conservative fields.

In mathematical terms, if \( abla \times \mathbf{F} = \mathbf{0} \), the field \( \mathbf{F} \) has no rotational component and is thus conservative, ensuring that any path between two points yields the same line integral value. Learning how to compute the curl is essential for diving into advanced topics of vector calculus and understanding intricate flow patterns.