Julia writes 2 fractions with the same denominator that have numerators 5 and 7 what could the denominator be if the sum is less
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Split up the boundary of <em>C</em> (which I denote ∂<em>C</em> throughout) into the parabolic segment from (1, 1) to (0, 0) (the part corresponding to <em>y</em> = <em>x</em> ²), and the line segment from (1, 1) to (0, 0) (the part of ∂<em>C</em> on the line <em>y</em> = <em>x</em>).
Parameterize these pieces respectively by
<em>r</em><em>(t)</em> = <em>x(t)</em> <em>i</em> + <em>y(t)</em> <em>j</em> = <em>t</em> <em>i</em> + <em>t</em> ² <em>j</em>
and
<em>s</em><em>(t)</em> = <em>x(t)</em> <em>i</em> + <em>y(t)</em> <em>j</em> = (1 - <em>t</em> ) <em>i</em> + (1 - <em>t</em> ) <em>j</em>
both with 0 ≤ <em>t</em> ≤ 1.
The circulation of <em>F</em> around ∂<em>C</em> is given by the line integral with respect to arc length,
where <em>T</em> denotes the <em>tangent</em> vector to ∂<em>C</em>. Split up the integral over each piece of ∂<em>C</em> :
• on the parabolic segment, we have
<em>T</em> = d<em>r</em>/d<em>t</em> = <em>i</em> + 2<em>t</em> <em>j</em>
• on the line segment,
<em>T</em> = d<em>s</em>/d<em>t</em> = -<em>i</em> - <em>j</em>
Then the circulation is
Alternatively, we can use Green's theorem to compute the circulation, as
The flux of <em>F</em> across ∂<em>C</em> is
where <em>N</em> is the <em>normal</em> vector to ∂<em>C</em>. While <em>T</em> = <em>x'(t)</em> <em>i</em> + <em>y'(t)</em> <em>j</em>, the normal vector is <em>N</em> = <em>y'(t)</em> <em>i</em> - <em>x'(t)</em> <em>j</em>.
• on the parabolic segment,
<em>N</em> = 2<em>t</em> <em>i</em> - <em>j</em>
• on the line segment,
<em>N</em> = - <em>i</em> + <em>j</em>
So the flux is