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
zepelin [54]
3 years ago
7

EXAMPLE 5 If F(x, y, z) = 4y2i + (8xy + 4e4z)j + 16ye4zk, find a function f such that ∇f = F. SOLUTION If there is such a functi

on f, then
Mathematics
1 answer:
Valentin [98]3 years ago
7 0

If there is such a scalar function <em>f</em>, then

\dfrac{\partial f}{\partial x}=4y^2

\dfrac{\partial f}{\partial y}=8xy+4e^{4z}

\dfrac{\partial f}{\partial z}=16ye^{4z}

Integrate both sides of the first equation with respect to <em>x</em> :

f(x,y,z)=4xy^2+g(y,z)

Differentiate both sides with respect to <em>y</em> :

\dfrac{\partial f}{\partial y}=8xy+4e^{4z}=8xy+\dfrac{\partial g}{\partial y}

\implies\dfrac{\partial g}{\partial y}=4e^{4z}

Integrate both sides with respect to <em>y</em> :

g(y,z)=4ye^{4z}+h(z)

Plug this into the equation above with <em>f</em> , then differentiate both sides with respect to <em>z</em> :

f(x,y,z)=4xy^2+4ye^{4z}+h(z)

\dfrac{\partial f}{\partial z}=16ye^{4z}=16ye^{4z}+\dfrac{\mathrm dh}{\mathrm dz}

\implies\dfrac{\mathrm dh}{\mathrm dz}=0

Integrate both sides with respect to <em>z</em> :

h(z)=C

So we end up with

\boxed{f(x,y,z)=4xy^2+4ye^{4z}+C}

You might be interested in
Which of the following is a coefficient in -11x + 9m + 7? There may be more
andreyandreev [35.5K]

Answer: -11 & 9

Step-by-step explanation: Coefficients are the numbers before the variables.

7 0
2 years ago
Help help help AAAAAAA math is hard
yuradex [85]

Answer: I believe it is h

Step-by-step explanation:

8 0
3 years ago
Read 2 more answers
What is the value of 5 in 3590​
Xelga [282]

Answer:

500

Step-by-step explanation:

The 5 means 500. The whole number is read three thousand-five hundred-ninety

7 0
2 years ago
On number 19 this needs to be shown full work I don’t know where to start can someone help me now ASAP!!!
DENIUS [597]

Answer:

1

sorry if I'm wrong, I'm in the 7th grade lol, did my best :) hope I sortof helped

Step-by-step explanation:

5-2=3

square root of 3 with 3 is 1

5 - 2 = 1 \\ 3 \sqrt{3 = 1}

7 0
2 years ago
(–9) • (8) – (9) • (–9)
elena-s [515]

Answer:

10

Step-by-step explanation:


7 0
3 years ago
Other questions:
  • The map of a walking trail is drawn on a coordinate grid with three points of interest the trail starts at R(-2,4) and goes to S
    7·2 answers
  • 5. The regular price of a Space Invader game is $52, but it is on sale. The discount is $13. What is percent discount is this?​
    6·1 answer
  • Find the mean absolute value of 0.4, 0.2, 0.4, 0.6​
    7·1 answer
  • How many terms does the polynomial x^2y^2 have?
    5·2 answers
  • A shipping box is 10 1/2 in wide, 16 1/2 in long, and 14 1/2 in high. What is the volume of the box in cubic inches?
    13·2 answers
  • If you eat four medium strawberries you get 48% of your daily recommended amount of vitamin C.what fraction of your daily amount
    8·1 answer
  • FIND TWO NUMBERS WHOSE SUM IS 21 AND DIFFERENCE IS 17<br><br>11 , 10<br>2 ,23<br>19 ,2​
    6·1 answer
  • Which represents the area of this rectangle?
    13·2 answers
  • -7x-17=2x+10<br> Show work
    7·1 answer
  • Help. will give brainliest. please hurry!!! &lt;3
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!