contestada

Consider f and c below. f(x, y) = x2y3i + x3y2j,
c.r(t) = t2 − 2t, t2 + 2t , 0 ≤ t ≤ 1 (a) find a function f such that f = ∇f. f(x, y) = (b) use part (a) to evaluate c f · dr along the given curve
c.

Relax

Respuesta :

LammettHash
If there is a scalar function [tex]f(x,y)[/tex] such that [tex]\mathbf f(x,y)=\nabla f(x,y)[/tex], then it would satisfy

[tex]\dfrac{\partial f}{\partial x}=x^2y^3[/tex]
[tex]\dfrac{\partial f}{\partial y}=x^3y^2[/tex]

From the first PDE, we have

[tex]f(x,y)=\dfrac13x^3y^3+g(y)[/tex]

Take the derivative with respect to [tex]y[/tex] to get

[tex]\dfrac{\partial f}{\partial y}=x^3y^2+\dfrac{\mathrm dg}{\mathrm dy}=x^3y^2[/tex]
[tex]\implies\dfrac{\mathrm dg}{\mathrm dy}=0\implies g(y)=C[/tex]

So,

[tex]f(x,y)=\dfrac13x^3y^3+C[/tex]


By the fundamental theorem of calculus,

[tex]\displaystyle\int_{\mathcal C}\mathbf f(x,y)\cdot\mathrm d\mathbf r=\int_{\mathcal C}\nabla f(x,y)\cdot\mathrm d\mathbf r=f(\mathbf r(1))-f(\mathbf r(0))[/tex]
[tex]=f(-1,3)-f(0,0)=-9[/tex]

Otras preguntas