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lutik1710 [3]
2 years ago
10

Consider the given vector field. F(x, y, z) = x2yz i + xy2z j + xyz2 k (a) Find the curl of the vector field. (b) Find the diver

gence of the vector field.
Mathematics
1 answer:
user100 [1]2 years ago
7 0

Answer:  The required answers are

(a)~curlF=\left(xz^2-xy^2,x^2y-yz^2,y^2z-x^2z\right),\\\\(b)~divF=2xyz+2xyz+2xyz=6xyz.

Step-by-step explanation:  We are given to find the curl and divergence of the following vector field :

F(x,y,z)=x^2yzi+xy^2zj+xyz^2k.

We know that, for a vector field F(x,y,z)=(F_1,F_2,F_3), we have

curlF=\left(\dfrac{\partial F_3}{\partial y}-\dfrac{\partial F_2}{\partial z},\dfrac{\partial F_1}{\partial z}-\dfrac{\partial F_3}{\partial x},\dfrac{\partial F_2}{\partial x}-\dfrac{\partial F_1}{\partial y}\right),\\\\\\divF=\dfrac{\partial F_1}{\partial x}+\dfrac{\partial F_2}{\partial y}+\dfrac{\partial F_3}{\partial z}.

The required partial derivatives are calculated as follows :

\dfrac{\partial F_1}{\partial x}=\dfrac{\partial}{\partial x}(x^2yz)=2xyz,\\\\\dfrac{\partial F_2}{\partial x}=\dfrac{\partial}{\partial x}(xy^2z)=y^2z,\\\\\dfrac{\partial F_3}{\partial x}=\dfrac{\partial}{\partial x}(xyz^2)=yz^2,\\\\\dfrac{\partial F_1}{\partial y}=\dfrac{\partial}{\partial y}(x^2yz)=x^2z,\\\\\dfrac{\partial F_2}{\partial y}=\dfrac{\partial}{\partial y}(xy^2z)=2xyz,\\\\\dfrac{\partial F_3}{\partial y}=\dfrac{\partial}{\partial y}(xyz^2)=xz^2,\\\\\dfrac{\partial F_1}{\partial z}=\dfrac{\partial}{\partial z}(x^2yz)=x^2y,\\\\\dfrac{\partial F_2}{\partial z}=\dfrac{\partial}{\partial z}(xy^2z)=xy^2,\\\\\dfrac{\partial F_3}{\partial z}=\dfrac{\partial}{\partial z}(xyz^2)=2xyz.

Therefore, we get

curlF=\left(xz^2-xy^2,x^2y-yz^2,y^2z-x^2z\right),\\\\divF=2xyz+2xyz+2xyz=6xyz.

Thus, the required answers are

(a)~curlF=\left(xz^2-xy^2,x^2y-yz^2,y^2z-x^2z\right),\\\\(b)~divF=2xyz+2xyz+2xyz=6xyz.

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For all real numbers a and b, 2a • b = a2 + b2 Is this true or false? Explain why it false or true.
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Answer:

It's false.... correct is a2+2ab+b2

Step-by-step explanation:

This cannot be factored anymore although. when we try to substitute a with 5 and b with 2, the answer in the right hand side of the equation is -9.

That's why it's false.

3 0
2 years ago
The sales data for January and February of a frozen yogurt shop are approximately normal. The mean daily sales for January was $
VashaNatasha [74]

Answer:

January had a higher z-score for sales on the 15th, and the value of that z-score was of 0.5.

Step-by-step explanation:

z-score:

In a set with mean \mu and standard deviation \sigma, the z-score of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

January:

The mean daily sales for January was $300 with a standard deviation of $20. On the 15th of January, the shop sold $310 of yogurt. This means, respectively, that \mu = 300, \sigma = 20, X = 310. So

Z = \frac{X - \mu}{\sigma}

Z = \frac{310 - 300}{20}

Z = 0.5

February:

The mean daily sales for February was $320 with a standard deviation of $50. On the 15th of February, the shop sold $340 of yogurt. This means, respectively, that \mu = 320, \sigma = 50, X = 340. So

Z = \frac{X - \mu}{\sigma}

Z = \frac{340 - 320}{50}

Z = 0.4

January had a higher z-score for sales on the 15th, and the value of that z-score was of 0.5.

3 0
2 years ago
15 points!!!
r-ruslan [8.4K]

Answer:

The correct option is 4.

Step-by-step explanation:

If a figure it dilated by scale factor k, then the image and preimage are similar figures and their corresponding sides are proportional.

In triangle I, the base of the triangle is 1 unit and length of the perpendicular is 1 units.

In triangle II, the base of the triangle is 1 unit and length of the perpendicular is 2 units.

\frac{1}{1}\neq \frac{1}{2}

The corresponding sides are not proportional. It means both triangles are not similar. So, there is no dilation transforming triangle I into triangle II.

Therefore the correct option is 4.

6 0
2 years ago
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