How do you solve the equation:

Mathematics

1 answer:

Molodets [167]3 years ago

5 0

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Consider F and C below. F(x, y) = x4y5i + x5y4j, C: r(t) = t3 − 2t, t3 + 2t , 0 ≤ t ≤ 1 (a) Find a function f such that F = ∇f.

vovikov84 [41]

$\mathbf f(x,y)=x^4y^5\,\mathbf i+x^5y^4\,\mathbf j$

We want a scalar function $f(x,y)$ whose gradient is equivalent to the vector field. That means

$\dfrac{\partial f}{\partial x}=x^4y^5\implies f(x,y)=\dfrac{x^5y^5}5+g(y)$
$\dfrac{\partial f}{\partial y}=x^5y^4\implies x^5y^4=x^5y^4+\dfrac{\mathrm dg}{\mathrm dy}$
$\implies\dfrac{\mathrm dg}{\mathrm dy}=0\implies g(y)=C$

So (a)

$f(x,y)=\dfrac{x^5y^5}5+C$

which means (b)

$\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=f(\mathbf r(1))-f(\mathbf r(0))=f(-1,3)-f(0,0)=-\dfrac{243}5-0=-\dfrac{243}5$

5 0

3 years ago

On a coordinate plane, a line goes through (negative 4, 2) and (negative 2, negative 4). Which are solutions of the linear equat

hammer [34]

Answer:

Y = (-3) *X - 10

solution for Y =0 ---> X = -10/3

Step-by-step explanation:

Two points:

(Y0 ; X0) = (-4 ; 2)

(Y1 ; X1) = (-2 ; -4)

There is only one line which pass for both of them, his general expresion is: