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
dedylja [7]
3 years ago
14

Show that the line integral is independent of path by finding a function f such that ?f = f. c 2xe?ydx (2y ? x2e?ydy, c is any p

ath from (1, 0 to (4, 1
Mathematics
1 answer:
Juli2301 [7.4K]3 years ago
8 0
I'm reading this as

\displaystyle\int_C2xe^{-y}\,\mathrm dx+(2y-x^2e^{-y})\,\mathrm dy

with \nabla f=(2xe^{-y},2y-x^2e^{-y}).

The value of the integral will be independent of the path if we can find a function f(x,y) that satisfies the gradient equation above.

You have

\begin{cases}\dfrac{\partial f}{\partial x}=2xe^{-y}\\\\\dfrac{\partial f}{\partial y}=2y-x^2e^{-y}\end{cases}

Integrate \dfrac{\partial f}{\partial x} with respect to x. You get

\displaystyle\int\dfrac{\partial f}{\partial x}\,\mathrm dx=\int2xe^{-y}\,\mathrm dx
f=x^2e^{-y}+g(y)

Differentiate with respect to y. You get

\dfrac{\partial f}{\partial y}=\dfrac{\partial}{\partial y}[x^2e^{-y}+g(y)]
2y-x^2e^{-y}=-x^2e^{-y}+g'(y)
2y=g'(y)

Integrate both sides with respect to y to arrive at

\displaystyle\int2y\,\mathrm dy=\int g'(y)\,\mathrm dy
y^2=g(y)+C
g(y)=y^2+C

So you have

f(x,y)=x^2e^{-y}+y^2+C

The gradient is continuous for all x,y, so the fundamental theorem of calculus applies, and so the value of the integral, regardless of the path taken, is

\displaystyle\int_C2xe^{-y}\,\mathrm dx+(2y-x^2e^{-y})\,\mathrm dy=f(4,1)-f(1,0)=\frac9e
You might be interested in
Line f has a slope of -3. what is the slope of the line that is parallel to line f
andre [41]

Answer:

A line parallel has the same slope, so -3

8 0
3 years ago
WILL GIVE BRAINLIEST What are the like terms in the algebraic expression? Negative a squared b + 6 a b minus 8 + 5 a squared b m
Minchanka [31]

Answer: B

Negative a squared b and 5 a squared b

Step-by-step explanation:

Given that:

Negative a squared b + 6 a b minus 8 + 5 a squared b minus 6 a minus b. That is,

- a^2b + 6ab - 8 + 5a^b - 6a - b

Collecting the like term by rearranging the expression

5a^2b - a^2b + 6ab - 6a - b

The like terms in the expression above are

5a^2b - a^2b.

The correct option is B:

Negative a squared b and 5 a squared b or (-a^2b and 5a^b)

4 0
3 years ago
Read 2 more answers
Is it possible for y to be an independent variable?????????
Ahat [919]
No. Hope this helps!
5 0
3 years ago
Read 2 more answers
Round 1.965 to the nearest tenth.
riadik2000 [5.3K]

Answer: 2

Step-by-step explanation:

5 0
3 years ago
HELP! Find the EXACT angle(s) on The Unit Circle (in radians) that would produce sinθ = −2√2
bazaltina [42]

Answer:

picture?

Step-by-step explanation:

4 0
3 years ago
Other questions:
  • Find the product of (3x+2)(3x+7)
    9·1 answer
  • Helpppppp me plssss ​
    7·1 answer
  • Simplify 4 to the 7th over 5 squared all raised to the 3rd power.
    13·1 answer
  • Complement and supplement of a 19
    15·1 answer
  • What is y + 9 < -6 ???
    8·1 answer
  • What is the value of 43 = 23? 2 7 8 16​
    5·1 answer
  • WILL GIVE BRAINLST <br><br> HAVE A AMAZING DAy !!
    14·2 answers
  • Anna bought a potted plant online. It cost $3.4 5plus 20% shipping and handling.What was the total cost
    12·1 answer
  • Please help I'm in fourth and I do not understand this at all :((
    6·2 answers
  • What is the smallest number that has both 6 and 9 as a
    13·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!