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Calculus of Multiple Variables - Tutorial with Problems, Solutions, MCQ Quiz- Part II : Functions of several variables ,Theorems, Coordinates



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In case you'd like to take a look at other tutorials we have, related to Calculus of multiple variables :


                                                                                                                                                                                                             
Functions of several variables ,Theorems and Coordinates

After you learn about this topic, you might benefit from these MCQ Quizzes:



Important points to remember :

Euler’s theorem : If a homogeneous function f(x,y)  of degree n exists then
1. x(∂f/∂x) + y(∂f/∂y) =nf(x,y)
2. x2(∂2f/∂2x) + 2xy(∂2f/∂x∂y) +  y2(∂2f/∂2y)  = n(n-1)f(x,y)

Taylor’s Theorem : Taylor’s  expansion of one variable can be extended to functions of two variables . If h,k are small
f(x+h,y+k) =f(x,y) +(h ∂/∂x+k ∂/∂y)f+(h ∂/∂x+k ∂/∂y)2f/2! +.......

Jacobians : The Jacobian matrix is the matrix of all first-order partial derivatives of a vector or scalar-valued function with respect to another vector. 

Lagrange’s method of  undetermined multipliers : Introducing how we use this to find the maximum or minimum value for a function.

Polar Coordinates :  It is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. 
Cylindrical Coordinates : It is an extension of polar coordinates, but we extend it into the third dimension as well.
Spherical  Coordinates : A spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers


Here are some of the problems in the tutorial which will help you apply the concepts introduced in the tutorials :


Easy  Problems :

1. Convert the equation r2cos 2(theta)=z from Cylindrical to Cartesian coordinates.
2. If  u=(x2+y2+z2)1/2 , prove that  2u/x2 +2u/y2 +2u/z2=0
3. Convert the equation x2-y2+2yz=25  to cylindrical coordinates
4. If z=log((x2+y2)/xy) , prove  that  2f/xy=2f/yx
5. Convert the equation 2x2+2y2-4z2=0 from Cartesian Coordinates  to Spherical coordinates.
6. Convert the equation ρ sin φ = 1  from spherical coordinates to Cartesian coordinates.
7. Find a Taylor Approximation to f(x, y) =(x+y+1)1/2     near the origin

8. Find the Taylor Polynomial of order 3 of f(x) = (x+1)1/2 
9.   Find  du/dx  if   u=x2y   and  x2+xy+y2=1

Moderately difficult Problems :

10. If  f(x,y)=0,  show  that  d2y/dx2 = -(q2r-2pqs+p2t) / q3 where  p= f/ x , q= f/ y ,  r= 2f/ x2, s= 2f/ x y , t= 2f / y2
11. Find  the  minimum  value  of  x2+y2+z2  subject to xyz=a3
12.  Find the  extreme  values  of  f(x,y,z)=2x+3y+z   such  that   x2+y2=5  and x+z=1
13.  Expand exlog(1+y) in power of x and y
14.  Find the shortest distance between the line  y=10-2x  and  the ellipse(x2/4)+(y2/9) =1 .
15. Find the minimum distance  from the point  (3,4,15)  to  the  cone   x2+y2=4z2
17.  Find the point on the plane 2x−3y+5z = 19 that is nearest to the  origin.

Difficult  Problems :

18.  Suppose the cost of manufacturing a particular type of box is such  that the base of the box costs three times as much per square foot as the sides and  top. Find the dimensions of the box that minimize the cost for a given volume.
19.  The cone z2=x2+y2 is cut by the plane  z = 1 + x + y  in some curve C. Find the point on C that is closest to the origin.
20.  Expand  x2y+3y-2   in   powers   of  (x-1)  and  (y+2)  using  Taylor’s  theorem
21.  Expand  exsin y  in  powers  of      x  and   y   as   far   as   terms   of      third   degree
22. Expand  exy  in  the  neighbourhood   of  (1,1)   up  to  second  degree  terms
23.  Let  f(x,y)  and  g(x,y)  be  two  homogeneous  functions  of  degree  m  and  n  respectively  where  m0.  Let  h=f+g .  If  x h/x  + y h/y =0,  show  that  f=g  for   some   scalar   
24.  If  u(x,y)=cos-1(x+y/(x +y)),  0<x, y<1,  Prove   that   x u/x  + y u/y= -1/2 cot u
25.  In   a  plane  triangle,  find  the  maximum  value  of  cosA cosB cosC

Complete Tutorial (there is an MCQ Quiz after this):




MCQ Quiz #1


Companion MCQ Quiz #1 for this topic- test how much you know about the topic. Your score will be e-mailed to you at the address you provide.

Google Spreadsheet Form


MCQ Quiz #2 for Multivariate Calculus- Tutorial 2

Read the questions in the document below. Fill up your answers in the answer submission form below it. Your score will be emailed to you.

MCQ Quiz: Multivariate Calculus- Tutorial 2


Answer Submission Form for MCQ Quiz #2

MCQ Quiz #2


In case you'd like to take a look at other tutorials we have, related to Calculus of multiple variables :