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ahjd2020
11.06.2021 •
Mathematics
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 =?
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Ответ:
It looks like the vector field is
F(x, y) = 3x ^(2/3) i + e ^(y/5) j
Find a scalar function f such that grad f = F :
∂f/∂x = 3x ^(2/3) => f(x, y) = 9/5 x ^(5/3) + g(y)
=> ∂f/∂y = e ^(y/5) = dg/dy => g(y) = 5e ^(y/5) + K
=> f(x, y) = 9/5 x ^(5/3) + 5e ^(y/5) + K
(where K is an arbitrary constant)
By the fundamental theorem, the integral of F over the given path is
∫c F • dr = f (0, 1) - f (1, 0) = 5e ^(1/5) - 34/5
Ответ:
Step-by-step explanation: