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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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The customers at the bank would like to withdraw money from the automated teller machine. Help them enter their personal ID numb
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To withdraw money from the automated teller machine the numeric pattern required to complete the personal ID numbers is 2, 13, 24, 35,46,57,68.

As given in the question,

Using automated teller machine customers at the bank withdraw money

Numeric pattern given for the personal ID numbers is

2, 13, 24, 35

First term 'a'= 2

Common difference between consecutive terms are same

'd'=13 -2                                                  

   =11

It is an arithmetic progression series

nth term = a+ (n-1)d

5th term = 2 +(5-1)11

               =46

6th term =2+(6 -1)11

              =57

7th term= 2+ (7-1)11

             = 68

Therefore, to withdraw money from the automated teller machine the numeric pattern required to complete the personal ID numbers is 2, 13, 24, 35,46,57,68.

The complete question is:

The customers at the bank would like to withdraw money from the automated teller machine. Help them enter their personal ID numbers by completing the numeric patterns. 2, 13, 24, 35, ___, ___, ____.

Learn more about pattern here

brainly.com/question/14720576

#SPJ1

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Answer:

Your answer is $16.40

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