Question

Show that the line integral ∮C fdx+gdy=∮ C (45x⁴y³dx+27y²x⁵dy)

is path-independent by applying the curl test. Compare the partial derivatives

∂y/∂f = ∂y/∂f and ∂x/∂g =− ∂g/∂x

Answer 1

The line integral ∮C (45x⁴y³dx + 27y²x⁵dy) is path-independent because the partial derivatives ∂f/∂y and ∂g/∂x are equal, satisfying the condition for the curl test confirming the presence of a potential function.

Step-by-step explanation:

The subject question asks to show that the line integral ∮C f dx + g dy = ∮ C (45x⁴y³dx + 27y²x⁵dy) is path-independent by applying the curl test. The curl test involves comparing the partial derivatives ∂f/∂y and ∂g/∂x to determine if they are equal. If they are equal, it implies that the scalar field has a potential function, and hence the line integral is independent of the path taken.

The given functions are f(x, y) = 45x⁴y³ and

g(x, y) = 27y²x⁵.

To apply the curl test, we find the partial derivatives:

∂f/∂y = ∂(45x⁴y³)/∂y

= 135x⁴y²

∂g/∂x = ∂(27y²x⁵)/∂x

= 135x⁴y²

As ∂f/∂y = ∂g/∂x, we can conclude that the given line integral is path-independent.