The vector field fx y 4x 3y 3 2xy5 i 3x 4y 2 5x 2y 4 j is...

The vector field F(x, y) = (4x^3y^3 - 2xy^5)i + (3x^4y^2 - 5x^2y^4)j is conservative. (a) Verify that Py = Qx. (b) Find a potential function for F(x, y). (c) Use your answer to part (b) to compute the line integral ∫C F·dr where C is the curve with parametrization C: r(t) = (t^2 + 1)i + (2t - 3)j, -1 ≤ t ≤ 2. (d) Describe all curves C which start at the origin and have ∫C F·dr = 0.

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00:01 Hi, so this is the given equation then we have to find the following.

00:04 So first let f is equal to pi plus qj and therefore we have p is equal to 4x cube y cube minus 2 x y to the power 5 and q is equal to 3x to the power 4 minus y square minus 5x square y to the 4.

00:24 Now py is equal to deba p by deba y which is equal to 12x cube y square minus 10 x y to the power 4 and q is equal to qy equal to deba q by deba y and that will be equal to 123x cube 12x cube y square minus 10 x y to the power 4 and therefore py equal to qx.

00:58 So f is conservative vector field.

01:01 Now moving to the question number b, f is a potential function of sp therefore f is equal to deba f that is 4x cube minus 4x cube y cube minus 4xy to the power 5 into i plus 3x to the power 4 y square minus 5x square y to the power 4 j is equal to deba f by deba xi plus deba f by deba y into j which implies that deba f by deba x is equal to 4x cube y cube minus 2xy to the power 5.

01:41 This is equation number one.

01:43 Similarly deba f by deba y equal to 3x to the power 4 y square minus 5x square y to the power 4.

01:52 This is equation number two.

01:53 Now on integration on both the side we get integration of deba f by deba x into dx equal to deba integration of 4x cube y cube minus 2xy to the power 5 into dx and therefore f is equal to x to the power 4 y cube minus x square y to the power 5 plus 5 of y.

02:21 This is equation number three.

02:23 Now from equation number three we have fy equal to deba f by deba y equal to 3x to the power 4 y square minus 5x square y to the power 4 plus 5y...

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