IGNOU BMTC 102 2026 - Check if the following integrals are independent of path and eva

Question

Check if the following integrals are independent of path and evaluate those which are independent.

$$\text{i) } \int_{(0,0)}^{(3,4)} (6xy - y^3) \, dx + (3x^2 - 3xy^2) \, dy$

$$\text{ii) } \int_{(-1,4)}^{(3,8)} (3x^2 - 2y^2) \, dx - 4xy \, dy$

09 Jan 2026

Answer :

Word Count : 390

(i) Consider the vector field: [ \mathbf{F}(x,y) = (P, Q) = (6xy - y^3, 3x^2 - 3xy^2) ] Check for path independence by verifying if (\frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y}). [ \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(3x^2 - 3xy^2) = 6x - 3y^2 ] [ \frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(6xy - y^3) = 6x - 3y^2 ] Since (\frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y}), the field is conservative, and the integral is independent of path. Now, find a potential function (f(x,y)) such that: [ f_x = P = 6xy - y^3, \quad f_y = Q = 3x^2 - 3xy^2 ] Integrate (P) w.r.t (x): [ f(x,y) = \int (6xy - y^3) , dx = 3x^2y - xy^3 + g(y) ] Differentiate

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