Which of the statements it true? Check all that apply If is a conservative vector field in R³ then there exists f such that ▼ƒ = F If is a conservative vector field then SF. dr will always yield the same value along the curve C from point A to point B regardless of the path taken 3 If is a conservative vector field in R³ then curl(F) = 0 Green's Theorem States: Let C be a positively orentied, piecewise-smooth, smple closed curve in the plane and let D be the region bound by C. If P and Q have continous partial derivatives on an open region that contains D, then Sc Pdx + Qdy = SSD(² or)dA ƏQ Ox
Which of the statements it true? Check all that apply If is a conservative vector field in R³ then there exists f such that ▼ƒ = F If is a conservative vector field then SF. dr will always yield the same value along the curve C from point A to point B regardless of the path taken 3 If is a conservative vector field in R³ then curl(F) = 0 Green's Theorem States: Let C be a positively orentied, piecewise-smooth, smple closed curve in the plane and let D be the region bound by C. If P and Q have continous partial derivatives on an open region that contains D, then Sc Pdx + Qdy = SSD(² or)dA ƏQ Ox
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Which of the statements is true?**
**Check all that apply**
- □ If \(\vec{F}\) is a conservative vector field in \(\mathbb{R}^3\) then there exists \(f\) such that \(\nabla f = \vec{F}\)
- □ If \(\vec{F}\) is a conservative vector field then \(\int_C \vec{F} \cdot d\vec{r}\) will always yield the same value along the curve \(C\) from point \(A\) to point \(B\) *regardless of the path taken*
- □ If \(\vec{F}\) is a conservative vector field in \(\mathbb{R}^3\) then \(\text{curl}(\vec{F}) = 0\)
- □ Green's Theorem States:
Let \(C\) be a positively oriented, piecewise-smooth, simple closed curve in the plane and let \(D\) be the region bound by \(C\). If \(P\) and \(Q\) have continuous partial derivatives on an open region that contains \(D\), then
\[
\int_C P \, dx + Q \, dy = \int_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA
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Transcribed Image Text:**Which of the statements is true?**
**Check all that apply**
- □ If \(\vec{F}\) is a conservative vector field in \(\mathbb{R}^3\) then there exists \(f\) such that \(\nabla f = \vec{F}\)
- □ If \(\vec{F}\) is a conservative vector field then \(\int_C \vec{F} \cdot d\vec{r}\) will always yield the same value along the curve \(C\) from point \(A\) to point \(B\) *regardless of the path taken*
- □ If \(\vec{F}\) is a conservative vector field in \(\mathbb{R}^3\) then \(\text{curl}(\vec{F}) = 0\)
- □ Green's Theorem States:
Let \(C\) be a positively oriented, piecewise-smooth, simple closed curve in the plane and let \(D\) be the region bound by \(C\). If \(P\) and \(Q\) have continuous partial derivatives on an open region that contains \(D\), then
\[
\int_C P \, dx + Q \, dy = \int_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA
\]
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