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hjlf
3 years ago
15

Use the Fundamental Theorem of Line Integrals to calculate ∫c F⃗ ⋅dr⃗ exactly, if F⃗ =3x2/3i⃗ +ey/5j⃗ , and C is the quarter of

the unit circle in the first quadrant, traced counterclockwise from (1,0) to (0,1).
∫c F⃗ ⋅dr⃗ =?
Mathematics
1 answer:
nignag [31]3 years ago
5 0

It looks like the vector field is

<em>F</em><em>(x, y)</em> = 3<em>x</em> ^(2/3) <em>i</em> + <em>e</em> ^(<em>y</em>/5) <em>j</em>

<em></em>

Find a scalar function <em>f</em> such that grad <em>f</em> = <em>F</em> :

∂<em>f</em>/∂<em>x</em> = 3<em>x</em> ^(2/3)   =>   <em>f(x, y)</em> = 9/5 <em>x</em> ^(5/3) + <em>g(y)</em>

=>   ∂<em>f</em>/∂<em>y</em> = <em>e</em> ^(<em>y</em>/5) = d<em>g</em>/d<em>y</em>   =>   <em>g(y)</em> = 5<em>e</em> ^(<em>y</em>/5) + <em>K</em>

=>   <em>f(x, y)</em> = 9/5 <em>x</em> ^(5/3) + 5<em>e</em> ^(<em>y</em>/5) + <em>K</em>

(where <em>K</em> is an arbitrary constant)

By the fundamental theorem, the integral of <em>F</em> over the given path is

∫<em>c</em> <em>F</em> • d<em>r</em> = <em>f</em> (0, 1) - <em>f</em> (1, 0) = 5<em>e</em> ^(1/5) - 34/5

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