1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Finger [1]
3 years ago
8

Show that if the vector field F = Pi + Qj + Rk is conservative and P, Q, R have continuous first-order partial derivatives, then

the following is true. ∂P ∂y = ∂Q ∂x ∂P ∂z = ∂R ∂x ∂Q ∂z = ∂R ∂y . Since F is conservative, there exists a function f such that F = ∇f, that is, P, Q, and R are defined as follows. (Enter your answers in the form fx, fy, fz.) P = Q = R = Since P, Q, and R have continuous first order partial derivatives, says that ∂P/∂y = fxy = fyx = ∂Q/∂x, ∂P/∂z = fxz = fzx = ∂R/∂x, and ∂Q/∂z = fyz = fzy = ∂R/∂y.
Mathematics
1 answer:
olchik [2.2K]3 years ago
4 0

Answer:

It is proved that \frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}, \frac{\partial P}{\partial z}=\frac{\partial R}{\partial x}, \frac{\partial Q}{\partial z}=\frac{\partial R}{\partial y}

Step-by-step explanation:

Given vector field,

F=P\uvec{i}+Q\uvec{j}+R\uvec{k}

Where,

P=f_x=\frac{\partial f}{\partial x}, Q=f_y=\frac{\partial f}{\partial y}, R=f_z=\frac{\partial f}{\partial z}

To show,

\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}, \frac{\partial P}{\partial z}=\frac{\partial R}{\partial x}, \frac{\partial Q}{\partial z}=\frac{\partial R}{\partial y}

Consider,

\frac{\partial P}{\partial y}=\frac{\partial}{\partial y}(\frac{\partial f}{\partial x})=\frac{\partial^2 f}{\partial y\partial x}=\frac{\partial^2 f}{\partial x\partial y}=\frac{\partial }{\partial x}(\frac{\partial f}{\partial y})=\frac{\partial Q}{\partial x}

\frac{\partial P}{\partial z}=\frac{\partial}{\partial z}(\frac{\partial f}{\partial x})=\frac{\partial^2 f}{\partial z\partial x}=\frac{\partial^2 f}{\partial x\partial z}=\frac{\partial }{\partial x}(\frac{\partial f}{\partial z})=\frac{\partial R}{\partial x}

\frac{\partial Q}{\partial z}=\frac{\partial}{\partial z}(\frac{\partial f}{\partial y})=\frac{\partial^2 f}{\partial z\partial y}=\frac{\partial^2 f}{\partial y\partial z}=\frac{\partial}{\partial y}(\frac{\partial f}{\partial z})=\frac{\partial R}{\partial y}

Hence proved.

You might be interested in
Please solve with explanation I’ve been asking all day (this is not a multiple choice question)
Anastaziya [24]

Answer:

a) SA = 522.9~cm^2

b) V_{cone} = 670.2~cm^3

c) V_{empty} = 1340.4~cm^3

Step-by-step explanation:

a)

For a cone,

SA = \pi r (L + r)

where L = slant height

L = \sqrt{r^2 + h^2}

We have r = 8 cm; h = 10 cm

L = \sqrt{(8~cm)^2 + (10~cm)^2}

L = \sqrt{164~cm^2}

SA = (\pi)(8~cm)(\sqrt{164~cm^2} + 8~cm)

SA = 522.9~cm^2

b)

V_{cone} = \dfrac{1}{3}\pi r^2 h

V_{cone} = \dfrac{1}{3}(\pi)(8~cm)^2(10~cm)

V_{cone} = 670.2~cm^3

c)

V_{cylinder} = \pi r^2 h

empty space = volume of cylinder - volume of cone

V_{empty} = V_{cylinder} - V_{cone}

V_{empty} = \pi r^2 h - \dfrac{1}{3}\pi r^2 h

V_{empty} = (\pi)(8~cm)^2(10~cm) - \dfrac{1}{3}(\pi)(8~cm)^2(10~cm)

V_{empty} = 1340.4~cm^3

3 0
3 years ago
What is the Value of the rational expression below when X is equal to 3?
Serhud [2]
First, you would have to plug 3 into the equation
15-(3)/9-(3)
Then do the math
12/6
This can be simplified, since 12 and 6 are divisible by 6, which is their greatest common factor
2/1
So the answer is D. 2

Hope this helps!
3 0
3 years ago
Find the surface area of the composite figure.
taurus [48]

Answer:

surface area is 39

Step-by-step explanation:

add the areas of each geometric figure making up the composite 3D figure.

first 3D figure

2+2+6+6

=16---eq 1

from third 3D figure

4+4+10+5

= 23

from 1 and 2

16+23

= 39

may be!! I'm not sure bout this answer

3 0
3 years ago
PLZZZ HELP XD WILL GIVE BRAINLIEST WHERE TO PUT THE POINT ON THE NUMBER LINE!!!!!
lana66690 [7]
Every number has its purpose in this case you start  at -3 then the negative sign tells you which way you are going but since we have another negative sign in between the number we go to the right instead of left with that said you put the point in number 4
8 0
3 years ago
The sides of a triangle are 7, 4, n. If n is an integer, state the largest and smallest possible values of n.
Sladkaya [172]

Answer:

4, 10

Step-by-step explanation:

The value for the third side of the triangle is given by

b-a < n < b+a where a and b are the two other sides of the triangle and b>a

7-4 < n < 7+4

3 < n < 11

Since n is an integer

4 would be the smallest value and 10 would be the largest

8 0
3 years ago
Read 2 more answers
Other questions:
  • A scuba diver starts at 85.6 meters below the surface and descends until he reaches 103.2 meters below sea level. How many meter
    6·1 answer
  • Help anyone? I've been trying to go through the lessons again and i can't figure this one out.
    6·1 answer
  • Mrs. Alford invested $6700 in securities. Part of the money was invested at 1% and part at 9%. The total annual income was $275.
    13·1 answer
  • 5/8 of a fence has been built. If there is still 40 feet to be built, how long will the fence be?
    7·1 answer
  • Sam has 13 1/2 yards of ribbon. He cuts them 3/4 yards . What is the maximum numbers of pieces of ribbon
    7·1 answer
  • Angie went shopping for her son’s graduation party. She spent $281 on food, $143 on paper goods, and $164 on decorations. What i
    7·1 answer
  • Plz Help your fav. Marshmello i wasn' t,,,,, xx born yesterday.
    7·2 answers
  • Math pls help???????
    11·2 answers
  • HELP PLEASE!!! 5-6! Help!!!
    6·1 answer
  • Our school planned to plant 150 seedlings in a tree planting activity. However,
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!