Chapter 13: Q22E (page 781)
Is the vector field shown in the figure conservative?
Explain.
The vector field is conservative
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Plot the gradient vector field of \(f\) together with a contour map of \(f\). Explain how they are related to each other. \(f(x,y) = \ln \left( {1 + {x^2} + 2{y^2}} \right)\)
\({\bf{F}}(x,y,z) = \left\langle {{x^2}, - y,z} \right\rangle \),
\(E\)is the solid cylinder y2+z2,,9,0,,x,,2
Let \({\rm{F}}\)be the vector field shown in the figure.
(a) If \({{\rm{C}}_{\rm{1}}}\)is the vertical line segment from\({\rm{( - 3, - 3)}}\)to\({\rm{( - 3,3)}}\), determine whether \(\int_{{{\rm{C}}_{\rm{1}}}} {\rm{F}} \cdot {\rm{dr}}\) is positive, negative, or zero.
(b) If \({{\rm{C}}_{\scriptstyle{\rm{2}}\atop\scriptstyle}}\)is the counterclockwise-oriented circle with a radius of 3 and center the origin, determine whether \(\int_{{{\rm{C}}_{\rm{2}}}} {\rm{F}} \cdot {\rm{dr}}\)is positive, negative, or zero.
Evaluate the line integral, where\({\rm{C}}\) is the given curve.
\({\rm{(0,0,0) to (1,2,3)}}\)\(\int_{\rm{C}} {\rm{x}} {{\rm{e}}^{{\rm{yz}}}}{\rm{ds}}\)Is the line segment from\({\rm{(0,0,0) to (1,2,3)}}\)
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