MTH301 Midterm Subjective Paper

Q01. Rectangular co-ordinates of a point are (

3,3, 3 3

a. Convert Rectangular co-ordinates to Spherical co-ordinates b. Convert Rectangular co-ordinates to Cylindrical co-ordinates. c. Verify your answer by converting back Rectangular co-ordinates from any one of these, that is, either from Spherical co-ordinates or Cylindrical co-ordinates.

Solution:

Rectangular co-ordinates ( , , ) x y z to Spherical co-ordinates ( , , ) ρ θ φ

x y z ρ =

tan y

x π θ θ

cos z Cos Cos x y z φ φ

45, Cos π

Rectangular co-ordinates ( , , ) x y z to Cylindrical co-ordinates ( , , ) r z θ

r x y

Rectangular · Co-ordinates · Spherical · Co-ordinates · Cylindrical · Co-ordinates

3,3, 3 3 45, ), 4 Cos π 18, , 3 3 π

Q02. Describe the set of all points in xyz-coordinate system at which f is continuous.

ln f x y z x y z

Solution:

ln

10, 0} f x y z x y z

D x y z x z

Q03. By considering different path approach, find whether

( 0 ,0 ) 3

lim

x y

x · x · y

exist or not.

(Note: In order to get full marks, do all necessary steps) Use chain rule to find r w and φ ∂w if z y x w where

θ φ θ φ θ

sin cos cos sin cos

r z r y r x

Solution:

To find r w , using chain rule, we have

r z

z w

r y

y w

r x

x w

r w

Substituting the values, we get

)(sin cos )(cos sin )(cos θ φ θ φ θ z y x r w

Putting the values of x, y and z, it yields

r r r r r r r w

sin cos sin cos (sin cos sin cos cos sin cos

θ θ θ φ φ θ θ φ θ φ θ

Similarly, for φ ∂w , using chain rule, we get

w w x w y w z

x y z φ φ φ φ