Chapter 13: Problem 16
Verify that the vector field is conservative. \(\mathbf{F}(x, y)=\frac{1}{x^{2}}(y \mathbf{i}-x \mathbf{j})\)
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In Exercises 19 and \(20,\) find the total mass of two turns of a spring with density \(\rho\) in the shape of the circular helix \(\mathbf{r}(t)=3 \cos t \mathbf{i}+3 \sin t \mathbf{j}+2 t \mathbf{k}\) \(\rho(x, y, z)=\frac{1}{2}\left(x^{2}+y^{2}+z^{2}\right)\)
Prove the property for vector fields \(\mathbf{F}\) and \(\mathbf{G}\) and scalar function \(f .\) (Assume that the required partial derivatives are continuous.) $$ \nabla \times(f \mathbf{F})=f(\nabla \times \mathbf{F})+(\nabla f) \times \mathbf{F} $$
Prove the property for vector fields \(\mathbf{F}\) and \(\mathbf{G}\) and scalar function \(f .\) (Assume that the required partial derivatives are continuous.) $$ \operatorname{div}(f \mathbf{F})=f \operatorname{div} \mathbf{F}+\nabla f \cdot \mathbf{F} $$
Evaluate the integral \(\int_{C}(2 x-y) d x+(x+3 y) d y\) along the path \(C\). \(C:\) arc on \(y=x^{3 / 2}\) from (0,0) to (4,8)
Evaluate \(\int_{C}\left(x^{2}+y^{2}\right) d s\) \(C:\) counterclockwise around the circle \(x^{2}+y^{2}=4\) from (2,0) to (0,2)
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