**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. x^{2}(∂^{2}f/∂^{2}x) + 2xy(∂^{2}f/∂x∂y) + y^{2}(∂^{2}f/∂^{2}y) = 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 r^{2}cos 2(theta)=z from Cylindrical to Cartesian coordinates.

2. If u=(x^{2}+y^{2}+z^{2})^{1/2} , prove that ∂^{2}u/∂x^{2} +∂^{2}u/∂y^{2} +∂^{2}u/∂z^{2}=0

3. Convert the equation x^{2}-y^{2}+2yz=25 to cylindrical coordinates

4. If z=log((x^{2}+y^{2})/xy) , prove that ∂^{2}f/∂x∂y=∂^{2}f/∂y∂x

5. Convert the equation 2x^{2}+2y^{2}-4z^{2}=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=x^{2}y and x^{2}+xy+y^{2}=1

Moderately difficult Problems :

10. If f(x,y)=0, show that d^{2}y/dx^{2} = -(q^{2}r-2pqs+p^{2}t) / q^{3} where p= ∂f/ ∂x , q= ∂f/ ∂y , r= ∂^{2}f/ ∂x^{2}, s= ∂^{2}f/ ∂x ∂y , t= ∂^{2}f / ∂y^{2}

11. Find the minimum value of x^{2}+y^{2}+z^{2} subject to xyz=a^{3}

12. Find the extreme values of f(x,y,z)=2x+3y+z such that x^{2}+y^{2}=5 and x+z=1

13. Expand e^{x}log(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 z^{2}=x^{2}+y^{2} 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 x^{2}y+3y-2 in powers of (x-1) and (y+2) using Taylor’s theorem

21. Expand e^{x}sin y in powers of x and y as far as terms of third degree

22. Expand e^{xy} 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):**

Multivariable Calculus Tutorial 2

**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.**

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**MCQ Quiz #2 for Multivariate Calculus- Tutorial 2**

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#### MCQ Quiz: Multivariate Calculus- Tutorial 2

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#### MCQ Quiz #2

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

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