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PtichkaEL [24]
3 years ago
8

Determine whether or not the vector field is conservative. if it is conservative, find a function f such that f = ∇f. (if the ve

ctor field is not conservative, enter dne.) f(x, y, z) = ye−xi + e−xj + 2zk
Mathematics
1 answer:
Akimi4 [234]3 years ago
8 0
A three-dimensional vector field is conservative if it is also irrotational, i.e. its curl is \mathbf 0. We have

\nabla\times\mathbf f(x,y,z)=-2e^{-x}\,\mathbf k

so this vector field is not conservative.

- - -

Another way of determining the same result: We want to find a scalar function f(x,y,z) such that its gradient is equal to the given vector field, \mathbf f(x,y,z):

\nabla f(x,y,z)=\mathbf f(x,y,z)

For this to happen, we need to satisfy

\begin{cases}f_x=ye^{-x}\\f_y=e^{-x}\\f_z=2z\end{cases}

From the first equation, integrating with respect to x yields

f_x=ye^{-x}\implies f(x,y,z)=-ye^{-x}+g(y,z)

Note that g *must* be a function of y,z only.

Now differentiate with respect to y and we have

f_y=-e^{-x}+g_y=e^{-x}\implies g_y=2e^{-x}\implies g(y,z)=2ye^{-x}+\cdots

but this contradicts the assumption that g(y,z) is independent of x. So, the scalar potential function does not exist, and therefore the vector field is not conservative.
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Using Cramer's rule to solve linear systems.
Rzqust [24]

Answer: Last Option

x=2,\ y=-5

Step-by-step explanation:

Cramer's rule says that given a system of equations of two variables x and y then:

x =\frac{Det(A_X)}{Det(A)}

y =\frac{Det(A_Y)}{Det(A)}

For this problem we know that:

Det(A) = |A|=\left|\begin{array}{ccc}4&-6\\8&-2\\\end{array}\right|

Solving we have:

|A|= 4*(-2) -(-6)*8\\\\|A|=40

Det(A_X) = |A_X|=\left|\begin{array}{ccc}38&-6\\26&-2\\\end{array}\right|

Solving we have:

|A_X|=38*(-2) - (-6)*26\\\\|A_X|=80

Det(A_Y) = |A_Y|=\left|\begin{array}{ccc}4&38\\8&26\\\end{array}\right|

Solving we have:

|A_Y|=4*(26) - (38)*8\\\\|A_Y|=-200

Finally

x =\frac{|A_X|}{|A|} = \frac{80}{40}\\\\x=2

y =\frac{|A_Y|}{|A|} = \frac{-200}{40}\\\\y=-5

4 0
3 years ago
Which individual right is found in both the Texas Bill of Rights and the United States Bill is Rights? A.) Rights to free speech
german

A.) Rights to free speech.

Rights for freedom of speech is the only one of these that is found in the US Bill of Rights, so therefore, that is your answer :)

6 0
3 years ago
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Identify the distance between the points (9,7,3) and (5,3,2), and identify the midpoint of the segment for which these are the e
Daniel [21]

Answer:

<em>dggkydkyrmhxmfjiildtlitslursusappurshfufzppufpu7sritstisitsuprspursur74s</em>

Step-by-step explanation:

she will is has p7w407wsdddfduufuur5e

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3 years ago
What is 48,100,000,000,000 in scientific notation? 4.81×10−11 4.81×10−13 4.81×1011 4.81×1013
SSSSS [86.1K]
Given number is 4,8100,000,000,000. 

In this number of zero is 11. So we can write it as 481 * 10^{11}

But for scientific notation we will enter decimal after two place in reverse direction.For this we will also multiply number by 10^2.

So number will be 4.81 * 10^{11} *10^2 = 4.81 * 10^{13}

So fourth option is correct.


4 0
3 years ago
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Write the equation of the line, given the y and x-intercepts.
Dimas [21]

Answer:

y = -(2/3)m - 2

Step-by-step explanation:

recall that the general equation of a straight line in slope-intercept form is

y = mx + b

where, m is the slope and b is the y - intercept

here we are given that the y - intercept is -2, hence the equation becomes:

y = mx + (-2)

or

y = mx -2

we are also given that the x-intercept is -3, which means that when y = 0, x = -3 (simply substitute this into our new equation to solve for m)

0 = m(-3) -2

0 = -3m - 2  (add 3m to both sides)

3m = -2  (divide both sides by 3)

m = -(2/3)

hence our equation is now

y = -(2/3)m - 2

5 0
3 years ago
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