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SVETLANKA909090 [29]
3 years ago
13

Help me pleaseeeeeee

Mathematics
1 answer:
Alenkinab [10]3 years ago
3 0

everything except the last two, I think.

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For each vector field f⃗ (x,y,z), compute the curl of f⃗ and, if possible, find a function f(x,y,z) so that f⃗ =∇f. if no such f
butalik [34]

\vec f(x,y,z)=(2yze^{2xyz}+4z^2\cos(xz^2))\,\vec\imath+2xze^{2xyz}\,\vec\jmath+(2xye^{2xyz}+8xz\cos(xz^2))\,\vec k

Let

\vec f=f_1\,\vec\imath+f_2\,\vec\jmath+f_3\,\vec k

The curl is

\nabla\cdot\vec f=(\partial_x\,\vec\imath+\partial_y\,\vec\jmath+\partial_z\,\vec k)\times(f_1\,\vec\imath+f_2\,\vec\jmath+f_3\,\vec k)

where \partial_\xi denotes the partial derivative operator with respect to \xi. Recall that

\vec\imath\times\vec\jmath=\vec k

\vec\jmath\times\vec k=\vec i

\vec k\times\vec\imath=\vec\jmath

and that for any two vectors \vec a and \vec b, \vec a\times\vec b=-\vec b\times\vec a, and \vec a\times\vec a=\vec0.

The cross product reduces to

\nabla\times\vec f=(\partial_yf_3-\partial_zf_2)\,\vec\imath+(\partial_xf_3-\partial_zf_1)\,\vec\jmath+(\partial_xf_2-\partial_yf_1)\,\vec k

When you compute the partial derivatives, you'll find that all the components reduce to 0 and

\nabla\times\vec f=\vec0

which means \vec f is indeed conservative and we can find f.

Integrate both sides of

\dfrac{\partial f}{\partial y}=2xze^{2xyz}

with respect to y and

\implies f(x,y,z)=e^{2xyz}+g(x,z)

Differentiate both sides with respect to x and

\dfrac{\partial f}{\partial x}=\dfrac{\partial(e^{2xyz})}{\partial x}+\dfrac{\partial g}{\partial x}

2yze^{2xyz}+4z^2\cos(xz^2)=2yze^{2xyz}+\dfrac{\partial g}{\partial x}

4z^2\cos(xz^2)=\dfrac{\partial g}{\partial x}

\implies g(x,z)=4\sin(xz^2)+h(z)

Now

f(x,y,z)=e^{2xyz}+4\sin(xz^2)+h(z)

and differentiating with respect to z gives

\dfrac{\partial f}{\partial z}=\dfrac{\partial(e^{2xyz}+4\sin(xz^2))}{\partial z}+\dfrac{\mathrm dh}{\mathrm dz}

2xye^{2xyz}+8xz\cos(xz^2)=2xye^{2xyz}+8xz\cos(xz^2)+\dfrac{\mathrm dh}{\mathrm dz}

\dfrac{\mathrm dh}{\mathrm dz}=0

\implies h(z)=C

for some constant C. So

f(x,y,z)=e^{2xyz}+4\sin(xz^2)+C

3 0
3 years ago
The value of the digit 5 in 25,513 is how many times the value of the digit 5 in 357
SOVA2 [1]

Are there any answer choices??

7 0
3 years ago
TrianglesABC and DEF are <br> Similar<br> Not similar
puteri [66]

Answer:

Similar

Step-by-step explanation:

We can say that the triangles are similar by looking at the angle measurements

Angle B = 35

Angle F = 55

Both angles A and D = 90 because these are right triangles

We can say the measure of angle E = 35 because the sum of angles in a triangle equal to 180

Same rule applies to angle C

4 0
2 years ago
How do you factor 4a^2+11a-20
bixtya [17]
You have to factor by grouping.
(4a - 5) (a + 4)
4 0
3 years ago
Please answer fast and give expaination I will give brainliest
loris [4]

Answer:

Agree.

Step-by-step explanation:

1. (5^r)^t can be represented simply as 5^rt.

2. when r as negative integer, t as positive integer, rt is always a negative integer

3. when ever the indices rt is negative, the number 5^rt will be less than 1, thus cannot be a whole number for sure.

5 0
2 years ago
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