Mathematics, 23.07.2019 21:00, ethangorrell67

Consider f and c below. f(x, y) = x2 i + y2 j c is the arc of the parabola y = 4x2 from (−1, 4) to (0, 0) (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.

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answered: ninaaforever

To answer this question, we need to make use of the concept of Potential Function (U)

f = grad U

The solution is:

U = x³/3 + y³/3 + K f = grad×U

b) - 54.41

f( x,y ) = x²×i + y²× j is a vectorial field that could be obtained from the gradient of an scalar, then

f( x, y ) = grad. U if that is true then:

dU/dx = x² ⇒ dU = x²× dx ⇒ Uₓ = x³/3 + g(y)

dU/dy = y² ⇒ according to that

g´(y) = y² ⇒ ∫g´(y) = y³/3 + K plugging that value in Uₓ we get the potential function U, or a family of functions according to constant (K)

U = x³/3 + y³/3 + K

b) To calculate ∫ grad.F * dr

C is the parabola y = 4×x²

we parametrize the curve as:

x = t then y = 4×t² r = ( t , 4×t² ) P ( -1 , 4) to Q ( 0 , 0) for - 1 ≤ t ≤ 0

If x = t dx = dt and dy = 8×t×dt so dr = ( 1 + 8×t ) dt

We choose K = 0 then U = x³/3 + y³/3

then U = ( t³/3 + 64×t⁶ )

∫₋₁⁰ ( t³/3 + 64×t⁶) × ( 1 + 8×t ) dt = ∫ [t³/3 + (8/3)×t⁴ + 64×t⁶ + 512×t⁷ ] ×dt

= t⁴/12 + (8/15)× t⁵ + (64/7) ×t⁷ + 512 /8× t⁸

= t⁴/12 + (8/15)× t⁵ + (64/7) ×t⁷ + 64 ×t⁸ | ₋₁⁰

= - [ 1/12 - 8/15 - 64/7 + 64 ]

= - 54.41

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answered: reganwade8298

We're looking for a scalar function f(x,y)

such that

$\nabla f(x,y)=\dfrac{\partial f}{\partial x}\,\mathbf i+\dfrac{\partial f}{\partial y}\,\mathbf j=x^2\,\mathbf i+y^2\,\mathbf j=\mathbf f(x,y)$

We have

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

So

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

By the fundamental theorem of calculus (for line integrals), we then have

$\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=\int_{\mathcal C}\nabla f\cdot\mathrm d\mathbf r=f(0,0)-f(-1,4)=0-21=-21$

answered: Guest

answer: all of the odd numbers were opened

the first person was the one that was opening all of the lockers. since one is odd, this causes all of the odd number lockers to be opened in the end.

answered: Guest

there are 70 more legs than eyes at the dog show (196 – 126 = 70). dogs have two more legs than eyes, and humans have the same number of legs as eyes. thus 70 is twice the number of dogs at the show, so the number of dogs is half of 70, which is 35. to get the number of humans, subtract 35 from 63: 63 – 35 = 28.

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