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Consider F and C below. F(x, y, z) = y2 sin(z) i + 2xy sin(z) j + xy2 cos(z) k C: r(t) = t2 i + sin(t) j + t k, 0 ≤ t ≤ π (a) Fi

Liono4ka [1.6K]

Answer:

a) $f (x,y,z)= xy^2\sin(z)$

b) $\int_C F \cdot dr =0$

Step-by-step explanation:

Recall that given a function f(x,y,z) then $\nabla f = (\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z})$ . To find f, we will assume it exists and then we will find its form by integration.

First assume that F = $\nabla f$ . This implies that

$\frac{\partial f}{\partial x} = y^2\sin(z)$ if we integrate with respect to x we get that

$f(x,y,z) = xy^2\sin(z) + g(y,z)$ for some function g(y,z). If we take the derivative of this equation with respect to y, we get

$\frac{\partial f}{\partial y} = 2xy\sin(z) + \frac{\partial g}{\partial y}$

This must be equal to the second component of F. Then

$2xy\sin(z) + \frac{\partial g}{\partial y}=2xy\sin(z)$

This implies that $\frac{\partial g}{\partial y}=0$ , which means that g depends on z only. So $f(x,y,z) = xy^2\sin(z) + g(z)$

Taking the derivative with respect to z and making it equal to the third component of F, we get

$xy^2\cos(z)+\frac{dg}{dz} = xy^2\cos(z)$

which implies that $\frac{dg}{dz}=0$ which means that g(z) = K, where K is a constant. So

$f (x,y,z)= xy^2\sin(z)$

b) To evaluate $\int_C F \cdot dr$ we can evaluate it by using f. We can calculate the value of f at the initial and final point of C and the subtract them as follows.

$\int_C F \cdot dr = f(r(\pi))-f(r(0))$

Recall that $r(\pi) = (\pi^2, 0, \pi)$ so $f(r(\pi)) = \pi^2\cdot 0 \cdot \sin(\pi) = 0$

Also $r(0) = (0, 0, 0)$ so $f(r(0)) = 0^2\cdot 0 \cdot \sin(0) = 0$

So $\int_C F \cdot dr =0$

5 0

3 years ago

How would you set up question 4?

Pavel [41]

Step-by-step explanation:

please mark me as brainlest

5 0

3 years ago

Read 2 more answers

1.)Each sandwich costs $.02 per cubic inch to make. What is the cost of each sandwich?

kirza4 [7]

Part 1) Each sandwich costs $.02 per cubic inch to make. What is the cost of each sandwich?

volume for cube sandwich is 125 in³
volume for pyramid sandwich is 41.67 in³

cost of each cube sandwich=0.02*125-----> $2.5

cost of each pyramid sandwich=0.02*41.67-----> $0.83

the answers Part 1) are
a) cost of each cube sandwich is $2.5
b) cost of each pyramid sandwich is $0.83

Part 2) If each sandwich sells for $4.00 and Mr. Sammie sells 100 Cube sandwiches and 100 Pyramid Sandwiches, how much profit would Mr. Sammie make?

we know that
Profit=Sells-Costs

step 1
find the sells
Sells=200*$4-----> $800

step 2
find the costs
Costs=100*$2.5+100*$0.83----> $333

step 3
find the profit
Profit=Sells-Costs
Profit=$800-$333-----> $467

the answer Part 2) is
the profit is $467

7 0

3 years ago

A partir del término general, calcular los 4 primerostérminos. (Considere que debe asignar

Scorpion4ik [409]

Answer:

$4,7,12,19,...$