PURE MATHEMATICS 212 Multivariable Calculus ASSIGNMENT 2 1. [2 marks] Name and sketch the surface: z...

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PURE MATHEMATICS 212 Multivariable Calculus ASSIGNMENT 2 1. [2 marks] Name and sketch the surface: z = 4x2 + y2 + 8x ?? 2y 2. [4 marks] (a) Find a parametric equation of the curve of intersection of the paraboloid z = 3 ?? x2 ?? y2 and the plane z = 2y. (b) Find an equation of the orthogonal projection of this curve to the xy-plane. 3. [4 marks] The curves below are given by their vector equations. Describe them in Cartesian coordinates. What are their geometric names? (a) r=(3 sin et) i + (3 cos et) j. (b) r = ??2 i + t j + (t2 ?? 1) k. 4. [3 marks] Dene the notion of a smooth curve.

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PURE MATHEMATICS 212 Multivariable Calculus ASSIGNMENT 2 2 2 1. [2 marks] Name and sketch the surface: z = 4x +y + 8x 2y 2. [4 marks] (a) Find a parametric equation of the curve of intersection of the paraboloid 2 2 z = 3x y and the plane z = 2y. (b) Find an equation of the orthogonal projection of this curve to the xy-plane. 3. [4 marks] The curves below are given by their vector equations. Describe them in Cartesian coordinates. What are their geometric names? t t 2 (a) r=(3 sine )i + (3 cose )j. (b) r =2i +tj + (t 1)k. 4. [3 marks] Dene the notion of a smooth curve. For which values of the parameter t is the curve 3 2 2 r =t cos(t)i + sin(t )j +t k smooth? Justify your answer. 5. [3 marks] Let u;v;w be dierentiable vector-valued functions of t. Prove that d du dv dw [u [vw)] = [vw] +u [ w] +u [v ] dt dt dt dt Hint: Apply the product laws for dot and cross product. R t 6. [4 marks] (a) Evaluate [(te )i + lntj]dt; (b) Find the arc length of the curve r(t) = (3 cost)i + (3 sint)j + 4tk; 0t 2.

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David · Accepted Answer

1) Given the surface, ? = 4?2 + ?2 + 8? − 2? below is the sketch of it. 
It can also be written as ? + 5 = 4(? + 1)2 + (? − 1)2
                          “An Elliptical Paraboloid”
2) A)  
Curve of intersection of ? = 3 − ?2 − ?2 &  ? = 2? is found by equating 
those two equations i.e. 3 − ?2 − ?2 = 2? which gives 
 ?2 + ?2 + 2? = 3 
Adding 1 on both sides and rearranging gives, 
(? − 0)2 + (? + 1)2 = 22 , which is a circle centred at (0,-1) and radius 2. 
Since ? = 2?, centre of the circle is (0, −1, −2) &  ?????? = 2. 
So, writing in parametric equation it will be  
"? = ? ???(?) , ? = −? + ? ???(?) , ? = −? + ? ???(?) , ? ∈ (?, ??)"
B) Orthogonal projection of this curve on XY plane is nothing but,  
(? − ?)? + (? + ?)? = ??, a circle centred at (0,-1) and radius 2.
3) A) Given ? = (3 sin(??))? + (3 cos(??))?  
This implies, ? = (3 sin(??)) & ? = (3 cos(??)),  clearly this parametric            
equation corresponds to a circle and its equation in Cartesian co-ordinates 
is     ?? + ?? = ?, a ??????
    B) Given ? = −2? + ?? + (?2 − 1)? 
This implies ? = −2 & ? = ? & ? = ?2 − 1 = ?2 − 1, thus the equation 
in   Cartesian co-ordinates is ? = ?? − ? & ? = −?,

PURE MATHEMATICS 212 Multivariable Calculus ASSIGNMENT 2 1. [2 marks] Name and sketch the surface: z = 4x2 + y2 + 8x ?? 2y 2. [4 marks] (a) Find a parametric equation of the curve of intersection of...

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