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Chapter 13: Problem 34

What does it mean that a line integral is independent of path? State the method for determining if a line integral is independent of path.

Short Answer

Expert verified
A line integral is independent of path when its value only depends on the endpoints of the path and not on the path itself. This occurs when the vector field we are integrating over is conservative, which means that the field is the gradient of another scalar field. To determine this, we check whether the curl of the field is the zero vector. This property applies only to fields in simply-connected domains, without holes.

Step by step solution

01

Understanding the concept of a line integral being independent of path

The statement that a line integral is independent of path means that the value of the integral over a field along a curve is the same, no matter what path we take between the two points. In other words, the only thing that matters for the integral's value are the endpoints of the path, not the specifics of the path taken itself.
02

Explaining the method to determine the path independence of a line integral

A line integral is independent of path if and only if the vector field being integrated is conservative. This correlates to the field being the gradient of some scalar field. The formal condition is that the curl of the vector field (\(\nabla\times\textbf{F}\)) is equal to the zero vector. If this is met, the field is conservative and hence, the line integral of the field is independent of path. Note that this applies only to vector fields in simply connected domains, which means those without holes.

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Most popular questions from this chapter

Prove the property for vector fields \(\mathbf{F}\) and \(\mathbf{G}\) and scalar function \(f .\) (Assume that the required partial derivatives are continuous.) $$ \nabla \times(f \mathbf{F})=f(\nabla \times \mathbf{F})+(\nabla f) \times \mathbf{F} $$

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) where \(C\) is represented by \(\mathbf{r}(t)\) \(\mathbf{F}(x, y)=x y \mathbf{i}+y \mathbf{j}\) \(\quad C: \mathbf{r}(t)=4 \cos t \mathbf{i}+4 \sin t \mathbf{j}, \quad 0 \leq t \leq \pi / 2\)

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) where \(C\) is represented by \(\mathbf{r}(t)\) \(\mathbf{F}(x, y, z)=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k}\) \(C: \mathbf{r}(t)=2 \sin t \mathbf{i}+2 \cos t \mathbf{j}+\frac{1}{2} t^{2} \mathbf{k}, \quad 0 \leq t \leq \pi\)

Let \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k},\) and let \(f(x, y, z)=\|\mathbf{F}(x, y, z)\| .\) $$ \text { Show that } \nabla(\ln f)=\frac{\mathbf{F}}{f^{2}} $$

Define the curl of a vector field.
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