Question

Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. \(\int_{C} \frac{2 x}{\left(x^{2}+y^{2}\right)^{2}} d x+\frac{2 y}{\left(x^{2}+y^{2}\right)^{2}} d y\) \(C:\) circle \((x-4)^{2}+(y-5)^{2}=9\) clockwise from (7,5) to (1,5)

Short answer

The evaluated value of the line integral is 0.

Step-by-step solution

Find a Potential Function

The given vector field is \(F = <\frac{2x}{\left(x^2+y^2\right)^2}, \frac{2y}{\left(x^2+y^2\right)^2}>\). In order to find a potential function f for F, solve the system of partial differential equations \(f_x = \frac{2x}{\left(x^2+y^2\right)^2}\) and \(f_y = \frac{2y}{\left(x^2+y^2\right)^2}\). This yields \(f(x, y) = -\frac{1}{x^2 + y^2}\).

Check if the Vector Field is Conservative

For a vector field to be conservative, the cross partials of the potential function must be equal (\(\frac{\partial f_x}{\partial y} = \frac{\partial f_y}{\partial x}\)). For the potential function derived in Step 1, the equality does hold, so the vector field is conservative.

Apply the Fundamental Theorem of Line Integrals

The Fundamental Theorem of Line Integrals states that for a conservative vector field F with potential function f, and a curve C parameterized by r(t) from t=a to t=b, the line integral ∫C F·dr = f(r(b)) - f(r(a)). Based on the given path from (7,5) to (1,5), substitute these values to evaluate the line integral. This yields -\(\frac{1}{9}\) - (-\(\frac{1}{9}\)) = 0 which is the result.