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Question 1: [15 marks]

A surface is defined by the following equation:

.

a) Find the equation of the tangent plane to the surface at the point P(3, –1). [10 marks]

b) Find the total differential of function z (x, y) at that point. [5 marks]

Question 2: [15 marks]

Find all critical points of the function and determine their character, that is whether there is a local maximum, local minimum, saddle point or none of these at each critical point.

Question 3: [20 marks]

A mass of a thin plate M is given by the double integral in the conditional units:

a) Determine the function describing the surface mass density. [3 marks]

b) Present a sketch of the plate and shade it (by hand or using technology). [7 marks]

c) Change the order of integration and calculate the plate mass M. [10 marks]

Question 4: [20 marks]

Find the work done by the force field

acting on a massive body to move it from point M to point N along the path given by the equation

;

where the parameter t varies from 0 to 1.

Present your answer in exact form and then calculate up to four significant figures.

Question 5: [25 marks]

Calculate the double integral in polar coordinates

over the domain D bounded in the first quadrant by the following lines:

- x = 0;

- y = x;

- x2 + y2 = 1/4.

Sketch the domain of integration. Present your answer in exact form and then calculate it up to two significant figures.

Question 6: [15 marks]

Find the total mass M of the wire, if the mass line density distribution is:

,

and the vector equation of the wire is r(t) = 4 cos t i + 4 sin t j + 3(t – 1) k, where 3/2 t 2 and the dimension is [|r|] = m. Present your answer in the dimensional exact form and then evaluate it approximately up to five significant figures.

Question 7: [25 marks]

i. Determine which of the vector fields given below is conservative:

ii. Show the points A(3, ) and B(4, –3 ) on the x, y-plane.

iii. Calculate the work done (in dimensionless units) by the conservative field between these two points. Present your answer in exact form and then calculate up to four significant figures.

Question 8: [15 marks]

For the given vector field calculate

(a) the divergence at the point P(15, 2, –3) and [5 marks]

(b) the curl at the same point P(15, 2, –3). [10 marks]

Determine, whether the vector field is solenoidal or conservative.