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]
2 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]2 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
How do I write the sum of 6x and 2x is at least 39 ?
tatiyna
<h2>Put x = \dfrac{39}{8} I write the sum of 6x and 2x is at least 39 .</h2>

Step-by-step explanation:

We have,,

6x and 2x

To find, the value of x = ?

According to question,

The sum of 6x and 2x is at least 39

∴ 6x + 2x = 39

⇒ 8x = 39

⇒  x = \dfrac{39}{8}

∴ The value of x = \dfrac{39}{8}

Thus, put x = \dfrac{39}{8} I write the sum of 6x and 2x is at least 39 .

5 0
3 years ago
Two similar circles are shown. The circumference of the larger circle, with radius OB, is 3 times the circumference of the small
zheka24 [161]

Answer:

The circumference of the smaller circle is:

C =  2*pi*x/3

Step-by-step explanation:

We know that for a circle of radius R the circumference is given by:

C = 2*pi*R

where pi = 3.14...

Here we have two circles, A and B, where B is the larger circle and A is the smaller circle.

We know that:

The circumference of B is 3 times the circumference of A.

The radius of circle B is: OB = x

The radius of circle A is: OA

We want to find an expression of OA.

The circumference of circle B will be:

C(B) = 2*pi*OB = 2*pi*x

The circumference of circle A will be:

C(A) = 2*pi*OA

And we know that the circumference of circle B is 3 times the circumference of circle A, then:

C(B) = 3*C(A)

replacing the equations for the circumferences, we get:

2*pi*x = 3*(2*pi*OA)

dividing both sides by 2*pi, we get:

x = 3*OA

Now we want to solve this for OA, then we need to isolate it,

x/3 = OA

We can conclude that the radius of the smaller circle is equal to x/3.

Then the circumference of circle A is:

C(A) = 2*pi*x/3

5 0
2 years ago
Almonds cost $15.50 for 6 pounds. Find the unit rate. Round to the nearest cent if necessary.
Salsk061 [2.6K]
15.50 for 6 lbs......15.50/6 =  $ 2.58 per lb
4 0
3 years ago
Jim is running on a trail that is 5/4 of a mile long. How many miles has he run so far?
vekshin1
1 mile and 1/4 of a mile
6 0
3 years ago
How to find the circumference of a circle that is 42m and has diameter
Nitella [24]

Answer:

Circumference of the circle is 131.88m

Step-by-step explanation:

we have to find the circumference of the given circle.

Given:

                     Diameter=42 m

          \\diameter=2*radius\\42 =2*radius\\radius=42/2\\radius=21 m

Circumference of a circle= 2\pi r

                          =2*\pi *21\\=2*3.14*21\\=42*3.14\\=131.88

So, the circumference of the circle is 131.88m

6 0
3 years ago
Other questions:
  • brenda savings account principal 850, 5.5% annual interest rate and is compounded quarterly for 1 year. what is the compound int
    5·1 answer
  • What is the slope of the line passing through (2,-1) and (1,2)
    8·1 answer
  • Which ordered pairs make the equation true?
    7·2 answers
  • 1) Identify the one number that comes next in the set: 5, 12..
    12·2 answers
  • What is 37% of 600? what are the steps.
    8·2 answers
  • Please help me!!!!!!​
    7·2 answers
  • Plz help me i need 100%
    13·2 answers
  • Help me please<br> i've ask this 3 times and didn't get a correct answer
    6·2 answers
  • What is the y-intercept of a line that has a slope of -3 and passes through point (0, -7)?
    12·1 answer
  • A train traveling 200 miles per hour for 1 1/2 hours covers the same distance as one traveling at 75 miles per hour for how many
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!