MTH301 Ass idea

Assignment No. 1 MTH 301 (Spring 2010)
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Assignment No. 1

MTH 301 (Spring 2010)

                                           

Maximum Marks: 25

                                                Due Date: 28^th April 2010  

Spherical co-ordinates of a point are?

Spherical co-ordinates of a point are .(4,3.14/2,3.14/6)
a. Convert spherical co-ordinates to Rectangular co-ordinates
b. Convert spherical co-ordinates to cylindrical co-ordinates.
c. Verify your answer by converting back spherical co-ordinates from any one of these, that is, either from rectangular co-ordinates or cylindrical co-ordinates.

Answer

Assuming the spherical coordinates are (r,theta,phi), rectangular coordinates are:
x=r sin(theta)cos(phi)=4 * sin(3.14/2) cos(3.14/6)=3.464
y=r sin(theta)sin(phi)=4 * sin(3.14/2) sin(3.14/6)=2
z=r cos(theta)=4 * cos(3.14/2)=0

Cylindrical coordinates are:
rho = r sin(theta)=4 * sin(3.14/2)=4
phi = phi=3.14/6
z= r cos (theta)=4 * cos(3.14/2)=0

To convert back from cylindrical coordinates
r=sqrt{rho^2+z^2}=rho=4
theta=atan2 {rho,z}=atan2(4,0)=3.14/2
phi=phi=3.14/6

http://www.cramster.com/answers-oct-09/calculus/please-help-calculus-lifesaver-rectangular-co-ordinates-of-a-point-are-a-convert_676169.aspx

Question Details:

Rectangular co-ordinates of a point are

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.

Response Details:

 

a)

So to go from Rec to Spherical you need to know that

Cartesian is in the form(x,y,z) and Spherical are in (r,j,q)

so to convert your x  your r is equal to  so (1^2)+(1^2)+(-2√2) which equals 10 then you sqrt that

and r =

then to find q use formula tanθ=y/x and solve for θ so  which equals 45 degrees

q=45 degrees

and to find j know that z = r*cos j so solve for j

(-2√2) = (√10 )* cos j

(-2√2)/(√10) = j =153.4349 degrees

so your complete new point is

b) To convert from rec to Cylindrical

Cartesian is (x,y,z) and Cyl. is (r,q,z) notice that they have z in common so your z remains the same no change

use x to find r with the equation

so (1^2)+(1^2) = 2 then sqrt that now       r=√2

find q using the same method as above which is the inversetan of y/x

θ=invtan(1/1)= 45    q=45

and z is the same. your new point is

c) I believe it says to only convert 1 backwards so to go from Cyn to Rec

z stays the same

x=rcosq

y=rsinq

x=√2 cos45=1

y=√2 sin45=1

so (1,1,-2√2)

http://www.cramster.com/answers-oct-09/calculus/calculus-please-help-lifesaver-describe-the-set-of-all-points-in-xyz-coordinate-system-at_675957.aspx

Question Details:

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

Answer :

 is continuous on the set of . is continuous on the set of . The product of two continuous functions is continuous on the intersections of their domains. Therefore,  is continuous on the set , which can further be described as

By considering different path approach, find whether?

By considering different path approach, find whether

Lim
(x,y)→(0,0)

xy
---------------
3x^2 + 2y^2

Exist or not ?

Answer

Substituting the limits to the function will give us 0 / 0, which is indeterminate: That is:

(0)(0) / [3(0)^2 + 2(0)^2] = 0 / 0

We need to use the Lhospital's rule considering two cases: For the limit of x and for the limit of y.. Thus, when we differentiate both the numerator and the denominator, we differentiate it using partial differentiation...

With respect to x (y is constant), the partial derivative of the function becomes:

y / [6x].

Substituting the limit, we get, 0 / 0. We apply LHospital's Rule (partial differentiation) again with respect to x (since we do the differentiation with respect to x from the start):


0 / 6 = 0. Take note of this 0.



Let's consider applying Lhospitals rule with respect to y. We have:


x / [4y]. Still 0 / 0, indeterminate. Applying the rule again wrt y:


= 0.

Thus, the limit exists.



To check on this answer, since our limit values are 0, lets remove the numerical coefficients:


xy / [x^2 + y^2].


Let x = y = 0.000000000001.

You will see that the denominator becomes very much larger than the numerator... A number divided by a very large number (almost infinity) is approximately equal to 0.

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