MATH 2013

April 1998


(20) 1. (a) Evaluate ${\displaystyle \int \int_R ydA}$, where R is the region in $\Bbb R^2$ between the curves y = x and y = x4, for $0 \leq x \leq 1$.
(b) Show that ${\displaystyle \int \int_{D_{a}} \frac{1}{x^2 + y^2 +
a^2}\;dA = \pi \ln(2)}$, where Da is the disc of radius a > 0, centred at the origin.
(c) Evaluate ${\displaystyle \int \int \int_E (x^2 + y^2 +
z^2)^{3/2} dV}$ where E is the region in $\Bbb R^2$ bounded by the spheres of radii a and b (b > a > 0), centred at the origin.
(10) 2. (a) Show that the integral ${\displaystyle I = \int
\int_S\;\frac{\sin^2 (\pi x)}{1 + 4(x+y)}\;dA}$ satisfies $0 \leq I \leq
1$, where S is the unit square $\{(x,y);\;0 \leq x \leq 1,\;0 \leq y
\leq 1\}$.
(b) Compute an approximate value for I by partitioning S by the lines $x = 1/2,\;\;y = 1/2$ and then using double sum with the integrand evaluated at the mid-points of the sub-regions defined by the partition.
(20) 3. Consider the vector field ${\displaystyle
\vec{F} = \left[ \begin{array}{c}
y\\ x-z\\ -y
\end{array} \right] = y \vec{i} + (x-z) \vec{j} + (-y) \vec{k}}$ defined on $\Bbb R^3$.
(a) Compute div$(\vec{F})$ and curl$(\vec{F})$.
(b) Find a potential function for $\vec{F}$.
(c) Evaluate the line integral ${\displaystyle \int^{\textstyle B}
_{{\textstyle A}_{\hspace*{0.1cm}{\textstyle \cal C}}}
\vec{F} \cdot d \vec{x}}$ where $A = (1,2,3),\;\;B = (3,2,1)$ and $\cal C$ is any smooth curve joining A to B.
(d) Compute the flux of $\vec{F}$ across the ellipsoid a2x2 + b2y2 + c2z2 = 1.
(e) Show that $\vec{F} = \mbox{curl}(\vec{G})$ where ${\displaystyle
\vec{G} = \left[ \begin{array}{c}
xz + \frac{1}{2}\;(y^2-z^2)\\
\end{array} \right] \;\;. }$
(10) 4. Use the change of coordinates in $\Bbb
R^2,\;\;(x,y) \rightarrow (u,v)$ defined by $u = x,\;\;v = x+y$, to show that ${\displaystyle \int \int_T e^{(x+y)^{2}} dA =
\frac{1}{2}\;(e-1)}$ where T is the triangular region in the (x,y) plane whose vertices are $(0,0),\;(1,0)$ and (0,1).
(15) 5. In this problem you may use the standard formulas for area of a disc and surface area of a sphere: $\pi R^2$ and $4 \pi R^2$, respectively, where R is the radius.
(a) Evaluate ${\displaystyle \oint_{\cal C} (2ydx + xdy)}$ where $\cal C$ is the circle of radius 3 centred at the origin, oriented clockwise.
(b) Compute the value of ${\displaystyle Q \equiv \int
\int_{\sum_{a}} \vec{E} \cdot d \vec{S}}$ where $
\sum_a$ is the sphere x2 + y2 + z2 = a2, oriented by its outward normal $\vec{N} = 2(x \vec{i} + y \vec{j} + z \vec{k})$ and $\vec{E} = e \vec{n}$, where e is a constant and $\vec{n} =
\vec{N}/\vert\vert\vec{N}\vert\vert$ is the unit normal. Express e in terms of Q and a.
(c) Suppose that two vector fields $\vec{H}$ and $\vec{J}$, defined and smooth on $\Bbb R^3$, are related by curl $(\vec{H}) = \vec{J}$. Show that the flux ${\displaystyle \int \int_{\sum} \vec{J} \cdot d
\vec{S}}$ of J through a surface $\Sigma$ is equal to the circulation ${\displaystyle \oint_{\cal C} \vec{H} \cdot d \vec{x}}$ of $\vec{H}$ around $\cal C$, where $\cal C$ is the boundary of $\Sigma$. [Assume that $\Sigma$ and $\cal C$ are smooth and compatibly oriented.]
(10) 6. (a) Determine which of the following sequences ${\displaystyle \{a_n\}_{n=1}^{\infty}}$ converges and obtain the limit in the convergent cases.
(i) ${\displaystyle a_n = 1 + \left( -\;\frac{1}{2} \right)^n\;\;;}$ (ii) ${\displaystyle a_n = \sin \left( \frac{n \pi}{2} \right)\;\;;}$
(b) Determine whether each of the following series is absolutely convergent , conditionally convergent or divergent.
(i) ${\displaystyle \sum_{n=1}^{\infty}\;\frac{1}{(n+3)}\;\;;}$ (ii) ${\displaystyle \sum_{n=0}^{\infty}\;\frac{(-1)^n}{n!}\;\;;}$ (iii) ${\displaystyle \sum_{n=0}^{\infty}\;\frac{(-1)^n}{2n+1}\;\;.}$
(15) 7. (a) Find either the general (n $^{\mbox{th}}$) term, or the terms of degree $\leq 5$, in the Taylor series expansion about x = 0 (Maclaurin series) for
(i) $\ln(1+x)$; (ii) ${\displaystyle \ln \left(
and determine the interval of convergence in each case.
(b) Set x = 1/3 in (a) (ii) and compute the approximation to $\ln(2)$ given by the sum of the first three non-zero terms of the Taylor series. Compute the error, if the exact value is $\ln(2) =
0.693147 ...$.
(c) Compute an approximation to ${\displaystyle F(x) = \int_0^x f(t)dt\;\;\mbox{where}\;\;f(t) =
\frac{e^t-1}{t}}$ by using the first three non-zero terms in the Taylor expansion for f(t) about t = 0. Use this result to approximate F(1/2).