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F=(2xy +z³)i + x³j + 3xz²k find a scalar potential and work done in moving an object in the field from (1,-2,1) to (3,1,4)

Alex73 [517]

Step-by-step explanation:

Given:

$\textbf{F} = (2xy + z^3)\hat{\textbf{i}} + x^3\hat{\textbf{j}} + 3xz^2\hat{\textbf{k}}$

This field will have a scalar potential $\varphi$ if it satisfies the condition $\nabla \times \textbf{F}=0$ . While the first x- and y- components of $\nabla \times \textbf{F}$ are satisfied, the z-component doesn't.

$(\nabla \times \textbf{F})_z = \left(\dfrac{\partial F_y}{\partial x} - \dfrac{\partial F_x}{\partial y} \right)$

$\:\:\:\:\:\:\:\:\: = 3x^2 - 2x \ne 0$

Therefore the field is nonconservative so it has no scalar potential. We can still calculate the work done by defining the position vector $\vec{\textbf{r}}$ as

$\vec{\textbf{r}} = x \hat{\textbf{i}} + y \hat{\textbf{j}} + z \hat{\textbf{k}}$

and its differential is

$\textbf{d} \vec{\textbf{r}} = dx \hat{\textbf{i}} + dy \hat{\textbf{j}} + dz \hat{\textbf{k}}$

The work done then is given by

$\displaystyle \oint_c \vec{\textbf{F}} • \textbf{d} \vec{\textbf{r}} = \int ((2xy + z^3)\hat{\textbf{i}} + x^3\hat{\textbf{j}} + 3xz^2\hat{\textbf{k}}) • (dx \hat{\textbf{i}} + dy \hat{\textbf{j}} + dz \hat{\textbf{k}})$

$\displaystyle = (x^2y + xz^3) + x^3y + xz^3|_{(1, -2, 1)}^{(3, 1, 4)}$

$= 422$

5 0

3 years ago

Vanessa bakes a rectangular cake that is 9 inches wide,13 inches long and 2 inches high. She removes the cake from the pan to fr

TEA [102]

This is a rectangular cake, and there are 3 measurements, meaning that you are evidently finding volume.

Let's go back, first off, the formula for volume is...

V = L x W x H (Length, Width, Height)
Your measurements are 9 in, 13 in, 2 in.

All you need to do is follow the formula and multiply them.

9 x 2 = 18
18 x 13 = 234 $in^{2}$

Your answer is C: 234 $in^{2}$

Hope this helps!

4 0

3 years ago

Suppose you begin a job with an annual salary of $32,900. Each year you are assured of a 5.5% raise. What its the total amount t

olga2289 [7]

Answer: the total amount that you can earn in 15 years is $737245. Option C

Step-by-step explanation:

You receive an annual salary of $32,900 and each year, you are assured of a 5.5% raise. Assuming there was no raise, you get 100% of your previous salary each year. With a raise of 5.5%, you will get 100 + 5.5 = 105.5% of your previous salary for each year. This is a geometric progression and we want to determine the sum of 15 terms(15 years).

The formula for the sum of terms in a geometric progression is

Sn = [a(r^n - 1)]/ r - 1

Sn = sum of n terms

a = the first term

n = number of terms

r = common ratio

From the information given,

a = 32900

n = 15

r = 105.5/100 = 1.055

S15 = [32900(1.055^15 - 1)] / 1.055 - 1

S15 = [32900(2.23247649 - 1)] / 0.055

S15 = 32900 × 1.23247649) / 0.055

S15 = 737245.0277

S15 = $737245

4 0

2 years ago

Trey’s desk is 3 feet wide. How many inches wide is the desk? A:3 inches B:24 inches C:36 inches or D: 48 inches

natali 33 [55]

Answer:

36

Step-by-step explanation:

6 0

2 years ago

Read 2 more answers

Write the complex number in polar form:

emmasim [6.3K]

9514 1404 393

Answer:

2√30 ∠-120°

Step-by-step explanation:

The modulus is ...

√((-√30)² +(-3√10)²) = √(30 +90) = √120 = 2√30

The argument is ...

arctan(-3√10/-√30) = arctan(√3) = -120° . . . . a 3rd-quadrant angle

The polar form of the number can be written as ...

(2√30)∠-120°

_____

<em>Additional comments</em>

Any of a number of other formats can be used, including ...

(2√30)cis(-120°)

(2√30; -120°)

(2√30; -2π/3)

2√30·e^(i4π/3)

Of course, the angle -120° (-2π/3 radians) is the same as 240° (4π/3 radians).

__

At least one app I use differentiates between (x, y) and (r; θ) by the use of a semicolon to separate the modulus and argument of polar form coordinates. I find that useful, as a pair of numbers (10.95, 4.19) by itself does not convey the fact that it represents polar coordinates. As you may have guessed, my personal preference is for the notation 10.95∠4.19. (The lack of a ° symbol indicates the angle is in radians.)

6 0

2 years ago