Which of the following statements is true about the differential form $(y^2-2xy)dx+(x^2-2xy)dy$ on the plane, and what is the value of the line integral of this form along any path from $(0,0)$ to $(1,1)$?

• A. The form is exact, and the integral from $(0,0)$ to $(1,1)$ is $0$.
• B. The form is not exact, and the integral from $(0,0)$ to $(1,1)$ is $1$.
• C. The form is not exact, and the integral from $(0,0)$ to $(1,1)$ is $-1$.
• D. The form is exact, and the integral from $(0,0)$ to $(1,1)$ is $2$.

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the problem. The differential form given is (y^2 - 2xy)dx + (x^2 - 2xy)dy. We need to determine whether this form is exact and calculate the line integral along any path from (0, 0) to (1, 1).
2. Step 2: Recall the definition of an exact differential form. A differential form Mdx + Ndy is exact if there exists a scalar function f(x, y) such that df = Mdx + Ndy. This implies that the partial derivatives of M and N must satisfy the equality ∂M/∂y = ∂N/∂x.
3. Step 3: Compute the partial derivatives. Let M = y^2 - 2xy and N = x^2 - 2xy. Calculate ∂M/∂y and ∂N/∂x. Use the formulas: ∂M/∂y = ∂(y^2 - 2xy)/∂y and ∂N/∂x = ∂(x^2 - 2xy)/∂x.
4. Step 4: Compare the partial derivatives. If ∂M/∂y = ∂N/∂x, the form is exact. If they are not equal, the form is not exact. This step determines whether the differential form is exact or not.
5. Step 5: If the form is exact, find the scalar function f(x, y) such that df = Mdx + Ndy. Then, evaluate f(x, y) at the endpoints (0, 0) and (1, 1) to compute the line integral. If the form is not exact, use other methods to evaluate the line integral along the given path.