Chapter 13: Q3TF (page 830)
If \({\rm{f}}\)has continuous partial derivatives of all orders on \({{\rm{R}}^{\rm{3}}}\),then\({\rm{div(curl}}\nabla {\rm{f) = 0}}\).
The given statement is true.
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Find the value of\(\iint_{H}{f}(x,y,z)dS \)..
Evaluate the line integral, where C is the given curve.
\(\) \(\int_{\rm{C}} {{{\rm{e}}^{\rm{x}}}} {\rm{dx}}\),C is the arc of the curve \({\rm{x = }}{{\rm{y}}^{\rm{3}}}\) from \({\rm{( - 1, - 1)}}\) to \({\rm{(1,1)}}\).
Use a calculator or CAS to evaluate the line integral correct to four decimal places.
\(\int_{\rm{c}} {{\rm{F}} \cdot {\rm{dr}}} \), where \({\rm{F}}\left( {{\rm{x,y}}} \right){\rm{ = xy i + siny j}}\)and \({\rm{r}}\left( {\rm{t}} \right){\rm{ = }}{{\rm{e}}^{\rm{t}}}{\rm{i + }}{{\rm{e}}^{{\rm{ - }}{{\rm{t}}^{\rm{2}}}}}{\rm{j}}\), \({\rm{1}} \le {\rm{t}} \le {\rm{2}}\)
Evaluate the line integral, where \({\rm{C}}\)is the given curve.
\(\int_{\rm{C}} {{{\rm{z}}^{\rm{2}}}} {\rm{dx + }}{{\rm{x}}^{\rm{2}}}{\rm{dy + }}{{\rm{y}}^{\rm{2}}}{\rm{dz}}\)\(C\) Is the line segment from?
Plot the gradient vector field of \(f\) together with a contour map of \(f\). Explain how they are related to each other. \(f(x,y) = \cos x - 2\sin y\)
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