Chapter 13: Problem 32
Describe an orientable surface
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In Exercises 73-76, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(C\) is given by \(x(t)=t, y(t)=t, 0 \leq t \leq 1,\) then \(\int_{C} x y d s=\int_{0}^{1} t^{2} d t\)
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 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)
In Exercises \(5-8,\) evaluate the line integral along the given path. \(\int_{C}(x-y) d s\) \(C: \mathbf{r}(t)=4 t \mathbf{i}+3 t \mathbf{j}\) \(0 \leq t \leq 2\)
Evaluate the integral \(\int_{C}(2 x-y) d x+(x+3 y) d y\) along the path \(C\). \(C:\) parabolic path \(x=t, y=2 t^{2}\) from (0,0) to (2,8)
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