Show that F is a conservative and use this fact to evaluate ∫C F · dr along the given curve.
13. F(x, y) = (4x3y2 – 2xy3) i + (2x4y – 3x2y2 + 4y3) j, C: r(t) = (t + sin πt) i + (2t + cos πt) j, 0 ⩽ t ⩽ 1
To show: F is a conservative vector field and evaluate the value of .
Consider the standard equation of curl F for
If the curl of a vector field F is zero, then F is said to be a conservative vector field.
The relation between the potential function f and vector field is , where .
Obtain the value of .
Substitute for , for and for in equation (3).
Since , F is a conservative vector and the domain of F is .
Thus, F is a conservative vector field.
Substitute for in .
Compare the equation (4) and equation (1).
Integrate equation (5) with respect to x.
Apply partial differentiation with respect to y on both sides of equation (7)
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