Come up with your own non-constant conservative vector field…

Come up with your own non-constant conservative vector field F→\style{font-size:35px}{\vec{F}}. Show that is it conservative. Then, find the work done by F→\style{font-size:35px}{\vec{F}} over the curve starting at (5,5)\style{font-size:35px}{(5,5)}, looping around the arrow on the x-\style{font-size:35px}{x-}axis, visiting Neptune, traveling to another universe, then coming back and ending up back at (5,5)\style{font-size:35px}{(5,5)}.Hint: work can be represented by ∫CF→∙ dr→\style{font-size:35px}{\int_C{\vec{F}\bullet\ d\vec{r}}}.
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Evaluate the integral ∫∫Q∫xz dV\style{font-size:35px}{\int\i…

Evaluate the integral ∫∫Q∫xz dV\style{font-size:35px}{\int\int\limits_Q\int{xz\ dV}} where Q\style{font-size:35px}{Q} is the quarter of the unit sphere x2+y2+z2=1\style{font-size:35px}{x^2+y^2+z^2=1} in octants I and IV (i.e. with 0≤x≤1\style{font-size:35px}{0\leq x\leq 1}, -1≤y≤1\style{font-size:35px}{-1\leq y\leq 1}, and 0≤z≤1\style{font-size:35px}{0\leq z\leq 1}).
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Evaluate ∫C(-y-2x) dx+(2x+8y) dy\style{font-size:35px}{\int_…

Evaluate ∫C(-y-2x) dx+(2x+8y) dy\style{font-size:35px}{\int_C{(-y-2x)\ dx + (2x+8y)\ dy}} where C\style{font-size:35px}{C} is the curve y=x2\style{font-size:35px}{y=x^2} from (0,0)\style{font-size:35px}{(0,0)} to (2,4)\style{font-size:35px}{(2,4)}, by first parametrizing C\style{font-size:35px}{C} as r→(t)\style{font-size:35px}{\vec{r}(t)}.
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Let SS be given by z=9-x2-y2z=9-x^2-y^2 with z≥0z\geq 0 with…

Let SS be given by z=9-x2-y2z=9-x^2-y^2 with z≥0z\geq 0 with CC as its counterclockwise boundary curve. Let F→(x,y,z)=\vec{F}(x,y,z)=.Use Stoke’s Theorem to find the line integral ∫CF→∙dr→\int_C{\vec{F}\bullet d\vec{r}} by first converting it to the flux integral as ∫S∫(∇×F→)∙N→ dS\int_S\int{(\nabla\times\vec{F})\bullet \vec{N}\ dS}.
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