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.