MTH301 Final Term Solved Subjective with References

MTH301 Final term Solved Subjective

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:

x y z to Spherical co-ordinates ( , , )

x y z

tan y

x

cos z

Cos Cos

x y z

45, Cos Use chain rule to find

r w

and  w if

z y x w where

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 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

Similarly, for  w , using chain rule, we get w w x w y w z

x y z

Substituting the values, it becomes

(2 )( cos cos ) (2 )( cos sin ) (2 )(0) w

x r y r z

Putting the values of x and y, it yields

sin cos cos cos sin cos r r w

Q02. Find the equation of tangent plane to the surface

ln y x z at the point P(-1,0,0) .

Solution:

The given surface is ln y x z which can be written as

)1( ln z y x z y x f