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trasher [3.6K]
2 years ago
14

Consider f and c below. f(x, y) = x2 i + y2 j c is the arc of the parabola y = 2x2 from (−1, 2) to (1, 2) (a) find a function f

such that f = ∇f. f(x, y) = x3 3​+ y3 3​ correct: your answer is correct. (b) use part (a) to evaluate c ∇f · dr along the given curve
c.
Mathematics
1 answer:
Korvikt [17]2 years ago
5 0
If there is some scalar function f(x,y) such that

\nabla f(x,y)=\mathbf f(x,y)=x^2\,\mathbf i+y^2\,\mathbf j

then we want to find f such that

\dfrac{\partial f}{\partial x}=x^2\implies f(x,y)=\dfrac{x^3}3+g(y)
\dfrac{\partial f}{\partial y}=y^2=\dfrac{\mathrm dg}{\mathrm dy}\implies g(y)=\dfrac{y^3}3+C
\implies f(x,y)=\dfrac{x^3}3+\dfrac{y^3}3+C

So the vector field \mathbf f(x,y) is conservative, which means the fundamental theorem applies; the line integral of \mathbf f along any path \mathcal C parameterized by some vector-valued function \mathbf r(t) over a\le t\le b is given by

\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=\int_{t=a}^{t=b}\mathbf f(\mathbf r(t))\cdot\dfrac{\mathrm d\mathbf r}{\mathrm dt}\,\mathrm dt=f(\mathbf r(b))-f(\mathbf r(a))

In this case,

\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=f(1,2)-f(-1,2)=\dfrac23
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