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Chapter 13: Problem 33

State the Fundamental Theorem of Line Integrals.

Short Answer

Expert verified
The Fundamental Theorem of Line Integrals states that if a particle moves in a vector field \( \mathbf{F} = \nabla f \) along a curve \( C \) from point \( A \) to point \( B \), the work done by the field on the particle is equal to the change in potential: \( \int_C \mathbf{F} \cdot d\mathbf{r} = f(B) - f(A) \).

Step by step solution

01

Understanding the Theorem

The Fundamental Theorem of Line Integrals states that if a particle moves in a vector field \( \mathbf{F} = \nabla f \) (where \( \nabla f \) is a gradient of a scalar function \( f \)) along a curve \( C \) from point \( A \) to point \( B \), the work done by the field on the particle is equal to the change in potential: \( \int_C \mathbf{F} \cdot d\mathbf{r} = f(B) - f(A) \). This equation shows the integral of the field \(\mathbf{F}\) over the curve \(C\) is equal to the difference in the values of the function \( f \) at points \( B \) and \( A \).

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Most popular questions from this chapter

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\mathbf{F}(x, y)=4 x \mathbf{i}-y^{2} \mathbf{j}\) and \((x, y)\) is on the positive \(y\) -axis, then the vector points in the negative \(y\) -direction.

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

Find \(\operatorname{div}(\operatorname{curl} \mathbf{F})=\nabla \cdot(\nabla \times \mathbf{F})\) \(\mathbf{F}(x, y, z)=x y z \mathbf{i}+y \mathbf{j}+z \mathbf{k}\)

Prove the property for vector fields \(\mathbf{F}\) and \(\mathbf{G}\) and scalar function \(f .\) (Assume that the required partial derivatives are continuous.) $$ \operatorname{div}(\operatorname{curl} \mathbf{F})=0 \quad \text { (Theorem } 13.3) $$

Define a line integral of a continuous vector field \(\mathbf{F}\) on a smooth curve \(C\). How do you evaluate the line integral as a definite integral?
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