Question 1. Write the dimensions of each of the following quantities in fundamental units (kg, meters, and seconds). Indicate if the quantity is dimensionless.
(a) Velocity.
(b) Angle (degrees or radians)
(c) Energy (joules)
(d) Π
(e) Force (Newtons)

In Einstein's Special Theory of Relativity, an important phenomenon is the Lorentz contraction. If L and L_o are both length in meters, indicate which of the following expressions for the Lorentz contraction is correct. v is velocity in m/s, and c is the speed of light in m/s.

(a) L = L_o(1/√(1-v²/c²))

(b) L = L_o.(1/√(c² - v²))

(c) L = L_o.(v²/√(c² - v²))

Briefly explain your answer.

Question 2. In class, the Helmholtz Theorem was presented in two different ways. State either one of these two definitions.

A vector field is given by

A^→(x, y, z) = x⁴ ∂_xy2zdy + zeta,

(a) Calculate the divergence of A. Show work!

(b) Calculate the gradient of the scalar function 0(x, y, z) = xy2 + x2y + z3. Show work!

(c) Calculate the magnitude and direction (as a unit vector) of maximum increase of fi(x, y, z) = xy2 + x2y + z3 at the point (1, 1, 1), including units. (Assume /3 is in volts.)

Question 3.

(a) Briefly define and draw an example of an irrotational field.

(b) Briefly define and draw an example of a solenoidal field.

(c) Draw an example of a field that is neither irrotational nor solenbidal.

(d) Draw an example of a field that is both irrotational or solenoidal.

Question 4. In rectangular coordinates, show that the curl of the vector field

A^→(x, y, z) = y²za^{^}_x - yz²a^{^}_y + z³a^{^}_z is solenoidal. Show work neatly!

Question 5. A point P(x, y, z) is shown in a rectangular coordinate system below. If P is expressed in spherical coordinates (r, θ, Φ), in the figure below show r, θ, and Φ, and draw the unit vectors a^{^}_r, a^{^}_θ and a^{^}Φ at P.

In rectangular coordinates a vector field is

V^→(x, y, z) = ya^{^}_x - xa^{^}_y

Convert this field to cylindrical coordinates V^→(Ρ, Φ, z). Neatly show all work.