MATHEMATICS FOR ENGINEERING ANALYSIS I 
 MATHEMATICA ASSIGNMENT II
									DUE DATE:    10/30/02

Needs["Graphics`Master`"] , Needs["Calculus`VectorAnalysis`"]  and SetCoordinates[Cartesian[x, y, z]] are 
needed for this assignment . The following are some of the commands you will need.
	 CrossProduct[u, v]  is  cross product of u and v .     u . v is the dot product of u and v. Sqrt[u . u] is the norm of u. PlotVectorField[{f, g}, {x, a, b}, {y, c, d}]; 
PlotVectorField3D[{f, g, h}, {x, a, b}, {y, c, d}, {z, 
                e, f }];  plot vector fields 
ParametricPlot[{x (t), y (t)}, {t, a, b}]   plots the curve parametrized by r(t)=x(t) i +y(t) j.     ParametricPlot3D[{x (t), y (t), 
                      z (t)}, {t, a, b}]   plots the curve parametrized by  r(t)=x(t) i +y(t) j+z(t)k
Grad[f] computes gradient. Div[{(Subscript[v, 1]), (Subscript[v, 2]), (Subscript[v, 3])}] is the divergence of [(Subscript[v, 1]), (Subscript[v, 2]), (Subscript[v, 3])]  Curl[{(Subscript[v, 1]), (Subscript[v, 2]), (Subscript[v, 3])}] is the curl of [(Subscript[v, 1]), (Subscript[v, 2]), (Subscript[v, 3])].

1. 	Let u =[(Subscript[u, 1]), (Subscript[u, 2]), 
          (Subscript[u, 3])]  ,  v=[(Subscript[v, 1]), (Subscript[v, 2]), (Subscript[v, 3])], and  w=[(Subscript[w, 1,]) (Subscript[w, 2]), (Subscript[w, 3])] be vectors in space. Show that:   (a) u.(v + w)=u.v + u.w	(b) u.(u × v)=

2.	For the given curve  C: r(t)  
	(a)   Sketch the graph of the curve over the given values of t.	(b)   Find the equation of the tangent line at the given point P. 
	(c)   Sketch the graphs of the curve and the tangent line.	(d)   Find the length of the curve.
	(e)   Find the curvature of the curve at P

	(i)    r(t) = 2(t-sin t) i + 2(1-cos t) j, -4π≤ t ≤ 4π    P = r((π/4))  
   	(ii)   r(t) = 2t i +  5cos t j + 5sin t k   -4π ≤ t ≤ 4π, P=( -8π,5,0)

3.	Draw the contour curve for the following functions
	a) f(x, y) = (x^2) - (y^2)			b) 
          f (x, y) = (1/12) (Sqrt[144 - 16  (x^2) - 9  (y^2)])
4.	Pot the graphs of the following functions. In each case use at least one option.(Note: the command Options[Plot3D] will give a list of all the options.) 

	a)	z = (e^(1 - (x^2) - (y^2)))				b)	z =   (e^(1 - (x^2) + (y^2)))			
	c)  	z=ln |y-(x^2)|			d)	z = cos ((((x^2) + 2  (y^2))/4))		
	e)	z=(1/12) (Sqrt[144 - 16  (x^2) - 9  (y^2)])		f)  	z=((5  (x^2) y)/(x^(2 + (y^2))))

5.	 Find the curl and divergence of the vector field  v(x,y,z)=sin x i+ cos y j -z k 
6.	Let u(x, y, z) =[z, x, y],   v(x,y,z) = [xy, yz, zx] and  f(x, y, z) = xyz. Find the following expressions.  (a)   curl(u  +  v)         (b)   curl(fu)        (c)  div(u  ×  v)        (d)   curl (grad(f))

7.. 	Let f(x, y) = ((((x - y))^2)/((x^2)  +  (y^2))).		Find  a) (Subscript[f, y]) (1,0)	b)  (Subscript[f, xxy])

8. 	 Evaluate the iterated integral
		a)	\(TextForm\`\(\(∫\_0\%2\)\(∫\_x\%2\( e\^\(-y\^2\)\) dy\ \ dx\)\(\ \ \ \ \ \)\)\)	b)	   \(TextForm\`\(\(∫\_0\%2\)\(∫\_x\%2\ \ \(\@\(16\  - \ x\^2\  - \ 
                  y\^2\)\) dy\ \ dx\)\(\ \ \ \ \ \)\)\)
9. 	Find the volume of the solid that lies below the surface z = 1 - xy and  above the plane region R bounded by y = x,  y = 1,  and x =0.
10. 	Evaluate the triple integral:	\(∫\_0\%\(\@2\)\(∫\_0\%\(\@\(2 - x\^2\)\)\(∫\_\(2\ x\^2\  + \ y\^\(\(2\)\(\ \ \ \ \)\)\)\%\(4 - y\^2\)y dz dy dx\)\)\)