prove the sufficiency, let L be any closed contour in region R occupied by field A, suppose curl A=0 everywhere in R. then, since R is simply connected. L is the boundary of some surface S lying entirely in R. By Stoke's theorem, integral of A =0. prove that A is a potential field. ( potential and irrotational fields). please in details thx:)

1-Determine the density of states for a two-dimensional continuous medium using periodic boundary conditions.

2- In the Einstein model, atoms are treated as independent oscillators. The Debye model, on the other hand, treats atoms as coupled oscillators vibrating collectively. However, the collective modes are regarded here as independent. Explain the meaning of this independence, and contrast it with that in the Einstein model.

Air at 10°C and I atm flows over a flat plate (30 cm x100 cm) at 20 m/s. The plate is maintained at 70°C. (a) calculate the boundary layer thikness at distances of 30 cm and 100 cm, (b) calculate the heat transfer from first 30 cm and from whole the plate.

• What is a challenge to reporting to more than one manager in a matrix organization? What might be a benefit?
• What do you think are the advantages and disadvantages of being employed by a boundary-less organization?
• What is the main cost of maintaining an organizational learning environment? What is the greatest benefit?
• Which kind of organization would you like to work for? Why?

Solve the following ordinary differential equations by separating variables and integration.
1. y'sinx=ylny, the boundary condition (b.c.) is that the function passes through (y=e,x=pi/3)
2. y'+2xy2=0, b.c. (y=1;x=2)
3. y'-xy=x, b.c. (y=1;x=0)

The data below shows the sugar content in grams of several brands of​ children's and​ adults' cereals. Create and interpret a​ 95% confidence interval for the difference in the mean sugar​ content, μC−μA. Be sure to check the necessary assumptions and conditions.​ (Note: Do not assume that the variances of the two data sets are​ equal.)

Children's cereal:

40.2, 59.4, 47.3, 43.1, 51.5, 48.4, 54.8, 44.3, 41.6, 42, 45.5, 42.7, 37.8, 59.9, 48, 54.1, 38.9, 55.5, 42.8, 34.9

​Adults' cereal: 23.7, 25.3, 2.5, 8.7, 2.4, 21.4, 16.2, 14.4, 23.1, 7.4, 5.9, 12.8, 16.3, 10.8, 1.3, 16.4, 2.2, 4.5, 2.6, 9.7, 12.2, 4.5, 4, 1.4, 6.9, 0.1, 18.8, 6.9, 19.1, 13

A) The confidence interval is (___,___) round to two decimal places

B) Based on these samples, with 95% confidence, children's cereals average between the lower boundary of ___ and upper boundary of ___ more grams of sugar content than adults cereals. (round to two decimal places).

Need the math explanation

1. The value of a weight vector is given as (w1=3, w2=-2, w0=1) for a linear model with soft threshold (sigmoid) function f(x). Define a decision boundary, where the values of the feature vector x result in f(x)=0.5. Plot the decision boundary in two dimensions.

2. Generating training samples: In two dimensional feature space x: (x1, x2,1), generate 20 random samples, for different values of (x1,x2), that belong to two different classes C1 (1) and C2 (0). The label of each feature vector is assigned so that the samples are linearly separable, i.e., can be separated by a linear model with a soft threshold (sigmoid) function. Plot the samples you generate in a two dimensional plane of (x1,x2). Hint: You may construct an underlying linear model to cut the plane in two halves. Then generate random samples at either side with proper labels.

3. Construct a quadratic error function using a learn model with a soft threshold (sigmoid) function for augmented feature vectors in n+1 dimensions. Derive a gradient decent algorithm for learning the weights. Write a program using either Matlab or Python to learn the weights using the training samples you generate from Prob. 2. Plot the resulting decision boundary.

4. Consider a linear combination of three radial basis functions. Draw a network structure for the model. Write a (pseudo) algorithm for learning the parameters of the model. (You determine what error function to use, what training samples to use, and write iterative equations for learning the parameters.)

Please show how you got to answer!

Snell's Law and the Law of Reflection explain how light is redirected when it encounters a surface between two media. In the extreme, light may only reflect at a boundary, and go back into the medium it was in. More often, some of it reflects and some goes through. If the boundary is plane and flat, then these laws are easy to interpret. When the boundary is curved, they describe happens at every point on the surface. One of the classic types of glass is called "crown" glass, which has an index of refraction for visible light of 1.52 and is usually free of significant impurities. It was one of the first glasses discovered, and windows are made from it. Another glass is called "flint" glass, and it has lead oxide added, which makes it heavier, more "dispersive", and increases its index of refraction to 1.62. 1. A ray of light enters a flat surface of crown glass at a 25 degree angle to the surface. At what angles do the reflected and refracted rays leave the surface? 2. As in the first part, but for flint glass, what are the angles? 3. For the flint glass, the refracted ray goes through the glass to the other side. If the glass is a parallel slab, what happens when the ray reaches the opposite side from the inside? At what angle to the surface does it exit the glass back into air? 4. What is the smallest angle to the surface that light can have and still be transmitted from the inside to the outside in the case of flint glass? What angle is the light going at as it leaves in that case? Hint: The laws of reflection and refraction are usually stated in terms of the angles to the perpendicular or "normal" to the surface. These questions are rephrased in terms of the angles to the surface so take care in interpreting the laws and your answers.

Snell's Law and the Law of Reflection explain how light is redirected when it encounters a surface between two media. In the extreme, light may only reflect at a boundary, and go back into the medium it was in. More often, some of it reflects and some goes through. If the boundary is plane and flat, then these laws are easy to interpret. When the boundary is curved, they describe happens at every point on the surface.

One of the classic types of glass is called "crown" glass, which has an index of refraction for visible light of 1.52 and is usually free of significant impurities. It was one of the first glasses discovered, and windows are made from it. Another glass is called "flint" glass, and it has lead oxide added, which makes it heavier, more "dispersive", and increases its index of refraction to 1.62.

1. A ray of light enters a flat surface of crown glass at a 25 degree angle to the surface. At what angles do the reflected and refracted rays leave the surface?

2. As in the first part, but for flint glass, what are the angles?

3. For the flint glass, the refracted ray goes through the glass to the other side. If the glass is a parallel slab, what happens when the ray reaches the opposite side from the inside? At what angle to the surface does it exit the glass back into air?

4. What is the smallest angle to the surface that light can have and still be transmitted from the inside to the outside in the case of flint glass? What angle is the light going at as it leaves in that case?

Hint: The laws of reflection and refraction are usually stated in terms of the angles to the perpendicular or "normal" to the surface. These questions are rephrased in terms of the angles to the surface so take care in interpreting the laws and your answers.

Iris is planning to make a charitable contribution to her church of $50,000. She can either give $50,000 of cash, or stock worth $50,000 and a basis of $10,000 and which she has held for 5 years. The donation of which property gives the better tax result? If Iris’ AGI was $90,000 what other considerations would she have to take into account?