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aliya0001 [1]
3 years ago
7

Let F ( x , y , z ) = ⟨ x − 1 z , y − 1 z , ln ( x y ) ⟩ . Verify that F = ∇ f , where f ( x , y , z ) = z ln ( x y ) . Evaluate

∫ C F ⋅ d r , where r ( t ) = ⟨ e t , e 2 t , t 2 ⟩ for 1 ≤ t ≤ 3 . Evaluate ∫ C F ⋅ d r for any path C from P = ( 1 2 , 4 , 2 ) to Q = ( 2 , 2 , 3 ) contained in the region x > 0 , y > 0 . In part (c), why is it necessary to specify that the path lies in the region where x and y are positive?
Mathematics
1 answer:
lesya692 [45]3 years ago
7 0

I suppose you mean

\vec F(x,y,z)=\left\langle\dfrac zx,\dfrac zy,\ln(xy)\right\rangle

Showing that f(x,y,z)=z\ln(xy) has gradient equal to \vec F is trivial.

By virtue of the existence of f, the gradient theorem applies here, so

\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\int_{(12,4,2)}^{(2,2,3)}\nabla f(x,y,z)\cdot\mathrm d\vec r=f(2,2,3)-f(12,4,2)=3\ln 4-2\ln48=\boxed{-\ln36}

The requirement that x>0 and y>0 is a consequence of the domain of the logarithm function. Both have to be positive in order for \ln(xy) to exist.

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In quadratic formula
lorasvet [3.4K]

x² - x - 12 = 0


x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}


a = 1, b = -1, c = -12

x = \dfrac{1 \pm \sqrt{(-1)^{2} - 4(1)(-12)}}{2(1)}


x = 4 or -3


Answer: 4 or -3

4 0
3 years ago
Simplify to create an equivalent expression of<br> 2(−n−3)−7(5+2n)
musickatia [10]
<h3>Answer:   -16n-41</h3>

Work Shown:

2(-n-3) -7(5+2n)

2(-n)+2(-3) - 7(5)-7(2n) ... distribute

-2n - 6 - 35 - 14n

(-2n-14n) + (-6-35)

-16n - 41

6 0
2 years ago
Given triangle ABC, A=105°, a=10cm, b=7cm, find B,C and c.
Whitepunk [10]

The value of the side c will be 5.56 cm and angle C is 32.49° and angle B is 42.51°.

<h3>What is law of cosine?</h3>

Let there is a triangle ABC such that |AB| = a units, |AC| = b units, and |BC| = c units and the internal angle A is of θ degrees, then we have:

a² + b² – 2ab cos C = c²

Given triangle ABC, A = 105°, a = 10 cm, b = 7 cm.

Then we have

7² + c² – (2 · 7 · c) cos 105° = 10²

c² + 3.62c – 51 = 0

On solving, we have

c = 5.56

Then the angle C will be

10² + 7² – 2 · 7 · 10 · cos C = 5.56²

                149 – 140 cos C = 30.91

                                 cos C = 0.8435

                                        C = 32.49°

We know that

       ∠A + ∠B + ∠C = 180°

105° + ∠B + 32.49° = 180°

                         ∠B = 42.51°

Learn more about law of cosines here:

brainly.com/question/17289163

#SPJ1

4 0
1 year ago
Uh please help 8x-4y=-4 <br> -3x+4y=5 <br> Subtract second equation from the first
crimeas [40]

Answer:

11x -8y = -9

Step-by-step explanation:

8x-4y=-4

-3x+4y=5

Subtract the second equation from the first

8x-4y=-4

-(-3x+4y=5)

-----------------------------

Distribute the minus sign

  8x-4y=-4

+(3x-4y=-5)

-----------------------------

11x -8y = -9

5 0
3 years ago
What is 20% of 9 no remanders or link
sergij07 [2.7K]

Answer:

1.8

Step-by-step explanation:

Your answer would be 1.8 because you would have to do 0.20x 9 and that gives you 1.8

Hope I was able to help!

4 0
2 years ago
Read 2 more answers
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