Mathematics: F(x, y) = x2 i + y2 j c is the arc of the parabola y = 2x2 from (1, 2) to (2, 8) (a) find a function f such that f = ∇f.

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F(x, y) = x2 i + y2 j c is the arc of the parabola

Ответ:

Lakenwilliams1250

12.03.2022

Mathematics

$\mathbf F(x,y)=x^2\,\mathbf i+y^2\,\mathbf j$

If a scalar function f(x,y)

exists for which $\nabla f(x,y)=\mathbf F(x,y)$ , then we'd have

$\dfrac{\partial f}{\partial x}=x^2\implies f(x,y)=\dfrac{x^3}3+g(y)$
$\implies\dfrac{\partial f}{\partial y}=y^2=g'(y)\implies g(y)=\dfrac{y^3}3+C$

so the scalar function would be given by

$f(x,y)=\dfrac{x^3+y^3}3+C$

where

is an arbitrary constant.

Presumably, this question is being asked in the context of the line integral of $\mathbf F(x,y)$ along the given contour

. Since $\mathbf F(x,y)$ is conservative - and consequently a scalar potential function f(x,y)

exists - we can simply use the gradient theorem to evaluate the line integral. We would get

$\displaystyle\int_C\mathbf F\cdot\mathrm d\mathbf r=f(2,8)-f(1,2)=\dfrac{511}3$

(and we get the same answer by parameterizing

and computing the integral as we usually would)

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F(x, y) = x2 i + y2 j c is the arc of the parabola y = 2x2 from (1, 2) to (2, 8) (a) find a function f such that f = ∇f.

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