Problem 28 - The Calculus with Analytic Geometry

Question

In Exercises 21 through 30, show that the value of the line integral is independent of the path and compute the value in any convenient manner. In each exercise, \(C\) is any sectionally smooth curve from the point \(A\) to the point \(B\). \(\int_{C}(y z+x) d x+(x z+y) d y+(x y+z) d z ; A\) is \((0,0,0)\) and \(B\) is \((1,1,1)\)

Step-by-Step Solution

Answer

The value is obtained by evaluating the potential function at the endpoints: \(\int_{C} \textbf{F} \cdot d \textbf{r} = f(1,1,1) - f(0,0,0)\).

Step 1: Verify the Independence of Path

To verify that the line integral is independent of the path, check if the vector field \(\textbf{F} = (y z + x, x z + y, x y + z)\) is conservative. This requires showing that the curl of \(\textbf{F}\) is zero: \(abla \times \textbf{F} = \textbf{0}\).

Step 2: Compute the Curl of \(\textbf{F}\)

Calculate the curl of \(\textbf{F} = (y z + x, x z + y, x y + z)\) using the definition of curl: \((\frac{\partial (x y + z)}{\partial y} - \frac{\partial (x z + y)}{\partial z}, \frac{\partial (y z + x)}{\partial z} - \frac{\partial (x y + z)}{\partial x}, \frac{\partial (x z + y)}{\partial x} - \frac{\partial (y z + x)}{\partial y})\). Confirm that each component is zero.

Step 3: Find a Potential Function

Since the curl of \(\textbf{F}\) is zero, find the potential function \(f\) such that \(\textbf{F} = abla f\). Find \(f\) by integrating each component. From \((\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}) = (y z + x, x z + y, x y + z)\), solve for \(f\).

Step 4: Compute the Potential Function

Integrate \(\frac{\partial f}{\partial x} = y z + x\) with respect to \(x\), yielding \(f(x, y, z) = \frac{x^2}{2} + x y z + g(y,z)\). Differentiate with respect to \(y\) and match with \(\frac{\partial f}{\partial y} = x z + y\) to find \(g(y,z)\). Continue this process for all coordinates.

Step 5: Evaluate the Potential Function at Endpoints

Once the potential function \(f(x, y, z)\) is found, evaluate it at the points \(A(0,0,0)\) and \(B(1,1,1)\). Use the Fundamental Theorem for Line Integrals: \(\int_{C} \textbf{F} \cdot d \textbf{r} = f(B) - f(A)\).

Step 6: Calculate the Line Integral

Evaluate the potential function \(f(x, y, z)\) at \(A(0,0,0)\) and \(B(1,1,1)\). For example, if \(f(x, y, z) = \frac{x^2}{2} + x y z + \frac{y^2}{2} + z y + \frac{z^2}{2}\), calculate \([f(1,1,1) - f(0,0,0)]\).

Key Concepts

potential function

A potential function is a scalar function whose gradient is equal to the given vector field. It's an important concept in vector calculus because it helps simplify the computation of line integrals. To find a potential function, you need to integrate the components of the vector field. In our solution, we verified that the vector field \(\textbf{F} = (y z + x, x z + y, x y + z)\) could be expressed as the gradient of some potential function \(f\). This means that \(f\) satisfies:

• \( \frac{\partial f}{\partial x} = y z + x \)
• \( \frac{\partial f}{\partial y} = x z + y \)
• \( \frac{\partial f}{\partial z} = x y + z \)

By integrating each component, you solve for \(f\). As an example, integrating \( \frac{\partial f}{\partial x} = y z + x \) with respect to \(x\) gives us part of the potential function, and we continue this process for \(y\) and \(z\).

curl of vector field

The curl of a vector field measures the field's rotation at a given point. Mathematically, it is a vector operation describing the infinitesimal rotation of a 3D vector field. For a vector field \(\textbf{F} = (P, Q, R)\), the curl is defined as:
\((abla \times \textbf{F}) = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\).

In our exercise, we calculated the curl of \(\textbf{F} = (y z + x, x z + y, x y + z)\) and showed that it was zero, proving that the vector field is conservative. This is crucial because a zero curl indicates that the vector field has no rotational component, confirming that the line integral is path-independent.

Fundamental Theorem for Line Integrals

The Fundamental Theorem for Line Integrals states that if a vector field \(\textbf{F}\) is conservative, then the line integral of \(\textbf{F}\) over a curve \(C\) from point \(A\) to point \(B\) can be evaluated by finding a potential function \(f\) such that \(\textbf{F} = abla f\):
\int_C \textbf{F} \cdot d\textbf{r} = f(B) - f(A)\.

This theorem simplifies the computation significantly because instead of evaluating the integral directly along the path, we can simply find and evaluate the potential function at the endpoints. In the given problem, we used this theorem effectively to compute the line integral.

conservative vector field

A vector field is called conservative if it can be expressed as the gradient of some scalar potential function. This implies that the field is path-independent – the line integral between two points depends only on the values of the potential function at the endpoints, not on the path taken. For a vector field \(\textbf{F}\) to be conservative, its curl must be zero everywhere in the region of interest.

• This means: \(abla \times \textbf{F} = \textbf{0}\).

Our vector field \(\textbf{F} = (y z + x, x z + y, x y + z)\) was shown to have a zero curl, thus proving it is conservative. This allowed us to find a potential function and apply the Fundamental Theorem for Line Integrals.

smooth curve

In line integrals, a smooth curve is one that is continuously differentiable, meaning it has no sharp corners or breaks. It ensures that the integral is well-defined and can be evaluated. For the integral \(\int_C \textbf{F} \cdot d\textbf{r}\) to be path-independent, \(C\) needs to be a smooth (or piecewise smooth) curve.

In our exercise, the given curve \(C\) is sectionally smooth, meaning it might be composed of several smooth segments. This is sufficient for applying the techniques used in solving the line integral problem as it allows for a clear and continuous evaluation along the curve's path.