SOME APPLICATIONS OF DIFFERENTIATION AND INTEGRATION

Area Under a Curve

The area under a curve: y = f(x) ³ 0 on [a, b], being a limit of elemental Riemann sum S f(x)D x, is given by:

A = ò _{(a,b)} f(x)dx.

Length of a Curve

The length of the curve y = f(x) on [a, b], being the limit of the elemental chordal sum

S D s = S Ö [(D x)^{2}+(f(x+D x)-f(x))^{2}] @ S Ö [1+(f¢ (x))^{2})]D x, is given by:

L = ò _{(a,b)} Ö [1+(f¢ (x))^{2}]dx.

Length of a Curve in Polar Coordinates

To a first order approximation the lenghth of the chord PQ,

|PQ| = Ö [r^{2}+(r+D r)^{2}-2r(r+D r)cos D q ]

= Ö [(2r^{2}+2rD r)(1-cos D q )+(D r)^{2})] @ Ö [r^{2}+(D r/D q )^{2}] D q .

Hence the length of the curve from q = a to q = b is given by

L(a , b ) = ò _{(}a ,b ) Ö [r^{2}+(dr/dq )^{2}] dq .

For an alternate derivation using the cartesian formula L = ò _{(a,b)} Ö [1+(dy/dx)^{2}]dx, x = r cos q , y = r sin q , dy/dq = (¶ y/¶ r)(dr/dq )+¶ y/¶ q = sin q dr/dq + r cos q , dx/dq = (¶ x/¶ r)(dr/dq )+¶ x/¶ q = cos q dr/dq - r sin q and so

L = ò _{(a,b)} Ö [(dy)^{2}+(dx)^{2}] = ò _{(}a ,b ) Ö [( sin q dr/dq + r cos q )^{2}+( cos q dr/dq - r sin q )^{2}]dq = ò _{(}a ,b ) Ö [r^{2}+(dr/dq )^{2}] dq . #

Area in Polar Coordinates

We find the area A(a , b ) of the sector enclosed between the curve r = r(q ) and the lines q = a , and q = b . To a first order approximation the area of a sector element OPQ is ½(rD q )r = ½r^{2}D q . Hence the required area being the limit of the Riemann sums S ½r^{2}D q is given by

A(a , b ) = ½ ò _{(a ,b )} r^{2}dq .

For alternate derivation based on the cartesian formula for the area bounded by a curve y = y(x) ³ 0, x Î [a, b], and the x-axis given by

A(a, b) = ò _{(a,b)} y(x) dx,

let q = a , b on the curve correspond to the points x = a, and x = b. Now,

area(sector OAB) = area(triangle ONB) –area(triangle OMA) - area(region MNBA)

= ½ r^{2}(b ) sin b cos b - ½ r^{2}(a) sin a cos a - ò _{(a,b)} y dx

= ½r^{2}(b ) sin b cos b - ½r^{2}(a) sin a cos a - ò _{(}a ,b ) (r sin q )(cos q dr/dq - r sin q ) dq

= - ò _{(}a ,b ) sin q cos q (rdr/dq )dq + ò _{(}a ,b ) r^{2} sin^{2} q dq + ½r^{2}(b )sin b cos b - ½r^{2}(a)sin a cos a

= - ½r^{2}sin q cos q |_{(}a ,b ) + ½ò _{(}a ,b ) r^{2}cos 2q dq + ò _{(}a ,b ) r^{2} sin^{2} q dq + ½r^{2}(b )sin b cos b -½r^{2}(a )sin a cos a

= ½ò _{(}a ,b ) r^{2 }dq . #

Center of Mass

For a set of discrete masses m_{i} placed at points (x_{i}, y_{i}, z_{i}), the x, y, and z-coordinates of the center of mass are

.

When the mass is distributed continuously along a curve, area, or volume the sums are to be replaced by appropriate integrals.

Center of Mass of a Curve

The x and the y-coordinates of the center of mass of the curve y = f(x), x Î [a, b], (regarded as a thin wire of of length L and of uniform thickness) are given by

= [ò _{(a,b)} x(ds/dx)dx]/ò _{(a,b)} (ds/dx)dx = [ò _{(a,b)} x Ö [1+(f¢ (x))^{2}]dx]/L,

= [ò _{(a,b)} f(x)(ds/dx)dx]/ò _{(a,b)} (ds/dx)dx = [ò _{(a,b)} f(x) Ö [1+(f¢ (x))^{2}]dx]/L.

Center of Mass of an Area

The x and the y-coordinates of the center of mass of the area A under the curve y = f(x) ³ 0, x Î [a, b] are

= [ò _{(a,b)} xf(x)dx]/ò _{(a,b)} f(x)dx = [ò _{(a,b)} xf(x)dx]/A,

= [ò _{(a,b)} [f(x)/2]f(x)dx]/ò _{(a,b)} f(x)dx = [(1/2) ò _{(a,b)} [f(x)]^{2}dx]/A.

Lateral Surface Area of a Frustrum of a cone

The lateral surface area A of a frustrum of top and bottom radii r and R and slant height H is given by:

A = p (R + r) H.

To derive it, cut the surface of the frustrum along the slant height and spread in a plane to get a sector of an annulus. If q is the angle subtended by the sector on the center and r the radius of the inner circle, r q = 2p r, (r +H)q = 2p R, so that q = 2p (R-r)/H, and hence A = (2p )^{-1}q [p ((r +H)^{2}-p r ^{2}] = (1/2)q ^{-1}Hq (2r +H) q = (1/2)(H/[2p (R-r)]) 2p (R-r)2p (R+r) = p H(R+r). #

Surface of Revolution

A line element of length D s with end points distant f(x) and f(x+D x) rotated about the x-axis generates the lateral surface of a frustrum of area p (f(x)+f(x+D x))D s @ 2p f(x)D s. Hence if a rectifiable curve y = f(x) ³ 0, x Î [a, b] is rotated about x-axis, the surface area generated, being the limit of the elemental sum S 2p f(x)D x, is given by

S_{rev} = ò _{(a,b)} 2p f(x)(ds/dx)dx = 2p ò _{(a,b)} f(x)Ö [1+(f¢ (x))^{2}]dx.

S_{rev} = 2p ´ L (Pappus theorem),

i.e., the surface of revolution equals the distance travelled by the center of mass of the curve multiplied by the length of the curve.

__ Example__. The surface of revolution of a circle of radius R whose center is at a distance H > R from the x-axis is given by S = 2p H´ 2p R.

__ Example__. The center of mass of a semi-circular arc of radius R is distant 4p R

Volume of Revolution

Let the continuous curve y = f(x) ³ 0 on [a, b] be rotated about the x-axis. The volume of the solid generated under the curve is given by: V_{rev} = p ò _{(a,b)} [f(x)]^{2} dx, which is obtained as the limit of the sum of volumes of the disks S _{i} p [f(x_{i})]^{2}(x_{i+1} – x_{i}), where x_{i}’s constitute a subdivision of the interval [a, b]. Clearly

V_{rev} = 2p ´ A (Pappus theorem),

i.e., the volume of revolution equals the distance travelled by the center of mass of area multiplied by the area.

__ Example__. The volume of revolution of a circle of radius R whose center is at a distance H > R from the x-axis is given by S = 2p H´ p R

__ Example__. The center of mass of a semi-circular disc of radius R is distant (4/3)p R

Volume by Slicing

Suppose a solid object has boundaries extending from x = a, to x = b, and that its cross-section in a plane passing through (x, 0, 0) and parallel to the yz-plane has area A(x). To a first order approximation, the volume of the slice of the object on the right to the plane of thickness D x is then A(x)D x so that the volume of the solid is the limit of the elemental sum S A(x)D x. This being a Riemann sum, the volume is given by the formula

V = ò _{(a,b)} A(x) dx.

The Stirling’s Formula for the Factorial and the Gamma Function

The Gamma function G (a), (a > 0) is defined by

G (a) = ò _{(0,¥ )} e^{-t}t^{a-1}dt.

The improper integral is convergent for all complex a with Rl a > 0. The Gamma function satisfies an important recurrence relation

G (a+1) = aG (a).

__ Proof__: Integrating by parts, G (a+1) = ò

Noting that G (1) = ò _{(0,}¥ ) e^{-t}dt = 1, it follows that G (2) = 1, G (3) = 2G (2) = 2!, G (4) = 3G (3) = 3!, … , so that in general

G (n+1) = n!, n = 0, 1, 2, 3, … .

Another useful constant is

G (1/2) = Ö p .

__ Proof__: Putting t = s

G (1/2) = ò _{(-¥ ,¥ )} exp[-x^{2}]dx.

To evaluate the integral, let I = ò _{(-}¥ ,¥ ) exp[-x^{2}]dx. Then

I^{2} = ò _{(-}¥ ,¥ ) exp[-x^{2}]dx ò _{(-}¥ ,¥ ) exp[-y^{2}]dy = ò _{(-}¥ ,¥ ) ò (-¥ ,¥ ) exp[-(x^{2}+y^{2})]dxdy.

Using the polar coordinates

I^{2} = ò _{(0,2}p ) ò _{(0,}¥ ) exp[-r^{2}]rdrdq = p ò _{(0,}¥ ) exp[-t]dt = p ,

so that I = Ö p . #

Next we derive the *Stirling’s formula* for the Gamma function:

G (a) @ (2p )^{½}e^{-a}a^{a-½}, a ® ¥ .

__ Proof__: Putting t = ax, G (a) = a

(e^{-a}a^{a-½})^{-1}a^{a} ò _{|x-1|}³ d e^{-ax}x^{a-1}dx £ a^{½ }ò^{ }_{|x-1|}³ d (e^{-x}x/e^{-1})^{a-1}e^{-x}dx = a^{ ½}(e^{-(1+}d )(1+d ))^{a-1} ® 0, a ® ¥ ,

where we have used:

[e^{-(1+}d )(1+d )]/[e^{-(1-}d )(1-d )] = [e^{-2}d (1+d )/(1-d )] = [1+2(d +d ^{2}+d ^{3}+…)]/[1+2d +(2d )^{2}/2!+ (2d )^{3}/3!+…] > 1.

Expanding the function ln x about the point x = 1, by the mean value theorem

ln x = (x-1) -[(x-1)^{2}/2!](1/x ^{2}),

for some x lying between 1 and x. Hence, with x = x _{x} Î (1-d , 1+d ),

(e^{-a}a^{a-½})^{-1}a^{a} ò _{|x-1|<}d e^{-ax}x^{a-1}dx = a^{ ½ }ò^{ }_{|x-1|<}d (e^{-(x-1)}x)^{a}x^{-1}dx = a^{ ½ }ò^{ }_{|x-1|<}d (e^{-(x-1)+lnx})^{a}x^{-1}dx

= a^{ ½ }ò _{|x-1|<}d exp[a{-(x-1)+(x-1)-((x-1)^{2}/2!)x ^{-2}}]x^{-1}dx

= a^{ ½ }ò _{|x-1|<}d exp[-a((x-1)^{2}/2!)x ^{-2}] x^{-1}dx.

Now,

a^{½ }ò _{|x-1|<}d exp[-a((x-1)^{2}/2)(1-d )^{-2}](1+d )^{-1}dx

£ a^{½ }ò _{|x-1|<}d exp[-a((x-1)^{2}/2!)x ^{-2}]x^{-1}dx

£ a^{½ }ò _{|x-1|<}d exp[-a((x-1)^{2}/2)(1+d )^{-2}](1-d )^{-1}dx

Putting x = 1 + (2/a)^{½}t(1+d ),

a^{½ }ò^{ }_{|x-1|<}d exp[-a((x-1)^{2}/2)(1+d )^{-2}](1-d )^{-1}dx = a^{½ }ò^{ }_{|t|<}d Ö (a/2)/(1+d ) exp[-t^{2}](1-d )^{-1}(2/a)^{½}(1+d )dt

@ [(1+d )/(1-d )]2^{½ }ò _{(}¥ ,¥ )^{ }exp[-t^{2}]dt

= [(1+d )/(1-d )]Ö (2p ).

Similarly,

a^{½ }ò _{|x-1|<}d exp[-a((x-1)^{2}/2)(1-d )^{-2}](1+d )^{-1}dx @ [(1-d )/(1+d )]Ö (2p ).

Hence,

[(1-d )/(1+d )]Ö (2p ) £ lim_{a}® ¥ (e^{-a}a^{a-½})^{-1}G (a) £ [(1+d )/(1-d )]Ö (2p ),

and letting d ® 0, we get lim_{a}® ¥ (e^{-a}a^{a-½})^{-1}G (a) = Ö (2p ), i.e., G (a) @ Ö (2p ) e^{-a}a^{a-½}, a ® ¥ , completing the proof of the Stirling’s formula. #

Clearly, the Stirling’s formula for the asymptotic evaluation of n! can be derived from the Stirling’s formula for the Gamma function. thus n! = G (n+1) = nG (n) @ n(2p )^{½}e^{-n}n^{n-½}, n ® ¥ , so that

n! @ (2p )^{½}e^{-n}n^{n+½}, n ® ¥ . (Stirling’s formula for n!)

An asymptotic evaluation of the semi-factorials m!! could be done by using:

2n!! º 2× 4× 6…(2n-2)× 2n = 2^{n}n! @ (2p )^{1/2}e^{-n}n^{n+1/2}2^{n}; (2n+1)!! º 1× 3× 5…(2n-1)× (2n+1) = (2n+1)!/(2n)!!.

Convergence of the Binomial Expansion on [-1, 1]

__ Theorem__. For a > 0, the binomial expansion

(1+x)^{a} = 1+S _{n³ 1} [a(a-1)(a-2)…(a-n+1)/n!]x^{n},

is valid for x Î [-1, 1], and, the convergence of the binomial series is uniform on [-1, 1].

We shall prove this result in due course of time. However, note that for the binomial series the limit ratio

lim_{n}® ¥ |a_{n+1}/a_{n}| = lim_{n}® ¥ |(a-n)/(a-n+1)x| = |x|,

so that the series converges for |x| < 1, and diverges for |x| > 1.

Note that the convergence of a Taylor’s series does not automatically imply that its sum equals the function value – for example, for the function f(x) defined by

for x ¹ 0, f¢ (x) = (2/x^{3})f(x) is a polynomial in 1/x multiplying exp{-1/x^{2}}. Differentiating this relation it follows that for all n ³ 1, f^{(n)}(x) is a polynomial in 1/x multiplying exp{-1/x^{2}}. Since for n ³ 0, x^{-n}exp{-1/x^{2}} = [x^{n}exp 1/x^{2}]^{-1} ® 0, x ® 0, it follows that f^{(n)}(0) = 0, n ³ 1. Hence each term in the Taylor’s expansion S _{n}³ 0 x^{n}f^{(n)}(0)/n! of f(x) about x = 0 is zero. Since the function is positive for each x ¹ 0, the Taylor’s expansion does not represent the value of the function.

Hence to prove that the Taylor’s expansion represents the function, it is necessary to show that the remainder R_{n}(x) tends to zero as n ® ¥ , where

f(x) = S _{0£ n£ m} (x-a)^{n}f^{(n)}(a)/n! + R_{m+1}(x).

Note that in a Taylor’s expansion about x = 0, the Cauchy’s form of remainder after n-terms is

R_{n}(x) = [f^{(n)}(x )/n!]x^{n},

where x = q x, with 0 < q < 1. Note that x = x (x, n), and, q = q (x, n). To analyze the binomial expansion for f(x) = (1+x)^{a}, let 0 £ m-1 < a < m, m ³ 1. As, f^{(n)}(x ) = [a(a-1)(a-2)…(a-n+1)](1+x )^{a-n},

|R_{n}(x)| = [a(a-1)(a-2)…(a-m+1)(-1)^{n-m}(m-a)… (n-a-2)(n-a-1)/n!] (1+x )^{a-n} x^{n}

= [a(a-1)(a-2)…(a-m+1)(-1)^{n-m}{G (n-a)/G (m-a)}/n!] (1+x )^{a-n} x^{n}

@ [a(a-1)(a-2)…(a-m+1)(-1)^{n-m}/G (m-a)]exp{a}(n-a)^{n-a-1/2}/(n^{n+1/2}] (1+x )^{a-n} x^{n}

= [a(a-1)(a-2)…(a-m+1)(-1)^{n-m}/G (m-a)]exp{a}(1-a/n)^{n-a-1/2}n^{a-2}] (1+x )^{a-n} x^{n}

@ [a(a-1)(a-2)…(a-m+1)(-1)^{n-m}/G (m-a)]n^{a-2} (1+x )^{a-n} x^{n}.

For whatever q Î (0, 1), and ½ > d > 0, |x/(1+q x)| < 1-d for x in [0, 1-d ], and for x in [-1/2 + d , 0], |x/(1+q x)| < (½ - d )/(1/2 + d ) , i.e., |x|/|1+q x| < max {1-d , (½ - d )/(1/2 + d )} provided x Î [-1/2+d , 1-d ]. Hence, this asymptotic estimate shows that R_{n}(x) ® 0, as n ® ¥ , if (i) –1/2 < x < 1, and (ii) if a < 2 and x = ½, 1. Moreover, the convergence is uniform for x Î [-1/2+d , 1-d ]. In the remaining cases because of unknown nature of x the estimate doesn’t say much. Hence there is a need for a better form of remainder in the Taylor’s expansion.

__ Taylor’s Expansion with an Integral form of Remainder__. Let f(x) be n+1 times continuously differentiable in the interior of an interval containing a and x and be continuous on the interval. Then,

f(x) = S _{0£ k£ n} f^{(k)}(a)(x-a)^{n}/n! + (n!)^{-1 ò }_{(a,x)}^{ }f^{(n+1)}(t)(x-t)^{n}dt.

__ Proof__: For n = 0, the formula reads f(x) = f(a) + ò

f(x) = S _{0}£ k£ n f^{(k)}(a)(x-a)^{n}/n! + f^{(n+1)}(a)(x-a)^{n+1}/(n+1)! + (1/(n+1)!) ò _{(a,x)}^{ }f^{(n+2)}(t)(x-t)^{n+1}dt,

which is the formula for the case ‘n+1’, completing the proof. #

__ Proof of the Convergence of the Binomial Expansion__: By the integral form of remainder in the binomial series

R_{n}(x) = [(n-1)!]^{-1} ò _{(0,x)} a(a-1)…(a-n+1)(1+t)^{a-n}(x-t)^{n}dt

= x^{n+1}[(n-1)!]^{-1} ò _{(0,1)} a(a-1)…(a-n+1)(1+xs)^{a}[(1-s)/(1+xs)]^{n}ds,

putting t = xs. For n > a, the magnitude of R_{n}(x) over the interval [-1, 1] is maximum when x = -1. This is because the factors |x|^{n+1} and (1+xs)^{a-n}, s Î (0, 1) both are maximum at x = -1 in the interval [-1, 1]. Hence

max_{x}Î [-1,1] |R_{n}(x)| = [(n-1)!]^{-1} ò _{(0,1)} |a(a-1)…(a-n+1)|(1-s)^{a}ds

= [a(a-1)…(a-m+1){G (n-a)/G (m-a)}/(n-1)!] ò _{(0,1)} (1-s)^{a}ds

@ [a(a-1)…(a-m+1)/G (m-a)]exp{a}(n-a)^{n-a-½ }n^{-n+½} ò _{(0,1)} [-(1-s)^{a+1}/(a+1)]_{(0,1)}

@ [a(a-1)…(a-m+1)/G (m-a)]n^{-a}(a+1)^{-1} ® 0, n ® ¥ . #

Note that the order of the rate of convergence of max_{x}Î [-1,1] |R_{n}(x)| is O(n^{-a}). Thus, we have proved that *the binomial series expansion* S _{n}³ 0 [a(a-1)(a-2)…(a-n+1)/n!]x^{n} *of* (1+x)^{a}, a > 0, c*onverges uniformly to the function* (1+x)^{a} *on the interval* [-1, 1]. Hence the applicability of the integral form of the remainder!

__ Corollary__. The function |x| can be uniformly approximated on [-1, 1] by a sequence of algebraic polynomials.

__ Proof__: In the binomial expansion |x| = Ö (x

Uniform Approximation of Functions in C[0, 1]

__ Theorem (Polygonal Approximation)__. If f(x) Î C[0, 1], given any e > 0 there exists a polygon W (x) such that |f(x) - W (x)| < e , for all x Î [0, 1].

__ Proof__: Since a continuous function on a closed interval is uniformly continuous, given an arbitrary e > 0, there exists a d > 0, such that |f(x)-f(y)| < e , whenever |x-y| < d . Take n > 1/d , define x

W _{n}(x) = [(x_{k+1}-x)/(x_{k+1}-x)]f(x_{k}) + [(x-x_{k})/(x_{k+1}-x)]f(x_{k+1}), x_{k} £ x £ x_{k+1}, 0 £ k < n..

Let x Î [0, 1]. We have a 0 £ k < n such that x_{k} £ x £ x_{k+1}. Then, clearly,

|f(x)-W _{n}(x)| £ (x_{k+1}-x)/ (x_{k+1}-x_{k})|f(x_{k})-f(x)| + (x-x_{k})/(x_{k+1}-x_{k})|f(x_{k+1})-f(x)| < e ,

i.e., |f(x)-W _{n}(x)| < e , x Î [0, 1], saying that f(x) is uniforly approximated by the polygons W _{n}(x). #

__ Theorem (Representation of Polygons)__. If W

W _{n}(x) = a_{0} + S_{1£ k£ n} a_{k}|x-x_{k}| + a_{n+1}x, x Î [0, 1].

__ Proof__: Without loss of generality we could assume that 0 = x

a_{0} + (x_{1}-x_{0)}× a_{1} + 0× a_{2} = W _{1}(0)

a_{0} + 0× a_{1} + x_{1}× a_{2} = W _{1}(x_{1})

a_{0} + (1-x_{1)}× a_{1} + 1× a_{2} = W _{1}(1),

which is true as the matrix of the system is

.

Now, the graph of W _{n}(x) on the interval [x_{n-1}, x_{n+1}] is a polygon with one break point. Let W _{1}(x) denote the polygon on [0, 1] obtained by extending this part on [0, 1] so that on [x_{0}, x_{n}] the graph is linear. Since W _{1}(x) has one break point on (0, 1), W _{1}(x) = a + bx + c|x-x_{n}|. Then the function W _{n-1}(x) = W _{n}(x)- W _{1}(x) is a polygon with break points x_{1}, x_{2}, … , x_{n-1} only. Accordingly, assuming the result for (n-1)-break points, it has the form W _{n}(x) - W _{1}(x) = W _{n-1}(x) = b_{0} + S _{1}£ k£ n-1 b_{k}|x-x_{k}| + b_{n}x, for some constants b_{k}’s. From this it follows that W _{n}(x) = W _{1}(x) + W _{n-1}(x) = a + bx + c|x-x_{n}| + b_{0} + S _{1}£ k£ n-1 b_{k}|x-x_{k}| + b_{n}x = a_{0} + S _{1}£ k£ n a_{k}|x-x_{k}| + a_{n+1}x, where a_{0} = a + b_{0}, a_{n+1} = b + b_{n}, a_{n} = c, and a_{k} = b_{k}, 1 £ k £ n-1, proving the result for n-break points. The theorem follows. #

__ Weierstrass’ Approximation Theorem__. If f(x) Î C[0, 1], given any e > 0 there exists an algebraic polynomial P(x) such that |f(x) - P(x)| < e , for all x Î [0, 1].

__ Proof__: Using the polygonal approximation theorem, f(x) can be uniformly approximated to within e /2 by a polygon W

Quadratic Surfaces

A general quadratic equation in the three space variables X, Y, Z is: AX^{2}+BY^{2}+CZ^{2} + 2HXY+2GXZ+2FYZ + LX+MY+NZ + C = 0. **If A, B, C, H, G, F are all zero** it reduces to a linear equation LX + MY + NZ + C = 0, which, if at least one of L, M, N is non-zero, represents **a plane** with the direction ratios of the normal to the pla being L, M, N. If also L, M, N are all zeros the equation represents **nothing**.

In the proper quadratic case when at least one of A, B, C, H, G, F is non-zero, the homogeneous second degree terms constitute a non-zero real quadratic form Q(X, Y, Z) = AX^{2}+BY^{2}+CZ^{2}+2HXY+2FYZ+2GXZ. The matrix

H =

of the quadratic form Q, being real symmetric, has real eign-values l , m , n and the associated orthonormal eigen-vectors u, v, w Î R ^{3}, so that [u | v | w]¢ H[u | v | w] = diag (l , m , n ), and with U = [u | v | w], Q(X, Y, Z) = [X Y Z]H[X Y Z]¢ = [X Y Z]U(U¢ HU)U¢ [X Y Z]¢ = l x^{2} + m y^{2} + n z^{2}, where (x, y, z) = (X Y Z)U are the coordinates of the point P = (X, Y, Z) with respect to the new orthogonal coordinate system with the axes in the direction of the unit vectors u, v, w. So in the new co-ordinate system the quadratic equation becomes

l x^{2} + m y^{2} + n z^{2} + lx + my + nz + c = 0.

For the coefficients of l , m , n there arise the follwing cases:

**(I)** **Only one of ****l , m , n is non-zero**, in which, without loss of generality, we can consider the case l , m = 0, n > 0. The equation then is: n z^{2} + lx + my + nz + c = 0. The term in z could be eliminated by a change of coordinate system: z ® z – ½n/n of taking the origin (0, 0, ½n/n ). Division by n leads to the equation z^{2} + l_{1}x + m_{1}y + c_{1} = 0.

(A) If l_{1}, m_{1} = 0, we get: **nothing**, if c_{1} > 0, **a single plane**, z = 0, if c_{1} = 0, and **a pair of parallel planes**, z = ± Ö (-c_{1}), if c_{1} < 0.

(B) If at least one of l_{1}, m_{1} is non-zero we change the coordinate system by shifting the origin to a point on the line l_{1}x + m_{1}y + c_{1} = 0 in the xy-plane through: x ® x+h, y ® y+k, where h and k are arbitrarily chosen so that l_{1}h + m_{1}y + c_{1} = 0. The equation now becomes z^{2} + l_{1}x + m_{1}y = 0. Next we rotate the xy-coordinate axes by an angle q so that the new x-axis lies in a dirction along the line l_{1}x + m_{1}y = 0 in the xy-plane through

,

i.e., x ® x cos q -y sin q , y ® x sin q + y cos q , so that l_{1}x + m_{1}y ® l_{1} (x cos q -y sin q ) + m_{1} (x sin q + y cos q ) = (l_{1} cos q + m_{1} sin q ) x + (-l_{1} sin q + m_{1} cos q ) y. Choosing: sin q = l_{1}/Ö (l_{1}^{2}+m_{1}^{2}), cos q = -m_{1}/Ö (l_{1}^{2}+m_{1}^{2}), the equation becomes 0 = z^{2} + l_{1} [x(-m_{1}/Ö (l_{1}^{2}+m_{1}^{2})–y(l_{1}/Ö (l_{1}^{2}+m_{1}^{2}))] + m_{1} [x(l_{1}/Ö (l_{1}^{2}+m_{1}^{2})+ y(-m_{1}/Ö (l_{1}^{2}+m_{1}^{2}))] = z^{2} - yÖ (l_{1}^{2}+m_{1}^{2}), which on putting c^{2} = Ö (l_{1}^{2}+m_{1}^{2}), gives: z^{2}/c^{2} = y, **a parabolic cylinder** with axis along the x-axis.

**(II) Two of l , m , n non-zero and one zero**, when without loss of generality, we take n = 0. Two cases arize:

**(A) l , m of the same sign**, when without loss of generality we can assume that l , m > 0. The equation is now l x^{2} + m y^{2} + lx + my + nz + c = 0. By a change of origin the term lx + my can be removed and we get the equation as: l x^{2} + m y^{2} + nz + c_{1} = 0.

**If n = 0, and c _{1} > 0 **we get

**If n = 0, and c _{1} = 0** we get l x

**If n = 0 and c _{1} < 0**, l x

**If n ¹ 0**, changing the origin at (0, 0, c/n) i.e., by z ® z-c/n, the equation becomes l x^{2} + m y^{2} + nz = 0, which is of one of the types: x^{2}/a^{2} + y^{2}/b^{2} = z, **an elliptical paraboloid** (upward), x^{2}/a^{2} + y^{2}/b^{2} = -z, another** elliptical paraboloid** (downward).

**(B) l , m of the opposite signs**, when without loss of generality we let l > 0, m < 0. From the equation l x^{2} + m y^{2} + lx + my + nz + c = 0, again, by a change of origin we remove the term lx + my to get: l x^{2} + m y^{2} + nz + c_{1} = 0.

**If n = 0**, the equation becomes l x^{2} + m y^{2} + c_{1} = 0. Then, **if c _{1} = 0**, the equation reduces to (y-mx)(y+mx) = 0, m = Ö (-l /m ) ¹ 0, representing

**If n ¹ 0**, the equation is: l x^{2} + m y^{2} + n(z + c_{1}/n) = 0, which by a change of origin to (0, 0, c_{1}/n) becomes l x^{2} + m y^{2} + nz = 0, which depending on the sign of n is of one of the types: x^{2}/a^{2} - y^{2}/b^{2} = z, **a hyperbolic paraboloid**, or, y^{2}/a^{2} - x^{2}/b^{2} = z, which is also a** hyperbolic paraboloid**.

**(III) All three of l , m , n non-zero**, in which case there arise two situations:

**(A)** **All of ****l , m , n have the same sign**, in which case, without loss of generality we can assume l , m , n > 0. Then, from the equation l x^{2} + m y^{2} + n z^{2} + lx + my + nz + c = 0 by a change of origin to (½l/l , ½m/m , ½n/n ), i.e., by the transformations: x ® x – ½l/l , y ® y – ½m/m , z ® z – ½n/n , the term lx + my + nz could be removed so that the equation becomes l x^{2} + m y^{2} + n z^{2} + c_{1} = 0. **If c _{1} > 0, nothing** is represented by the equation.

**(B) Two of l , m , n have the same sign and one an opposite sign**, when without loss of generality we can assume l , m > 0, n < 0. Then, as before, by the change of origin to (½l/l , ½m/m , ½n/n ), the term lx + my + nz could be removed and the equation becomes l x^{2} + m y^{2} + n z^{2} + c_{1} = 0.

**If c _{1} = 0**, the equation can be put in the standard form: x

**If c _{1} > 0**, the equation can be put in the form: x

**If c _{1} < 0**, the equation can be put in the form: z

Hence we have proved that a quadratic equation in x, y, z could represent one of the following:

(1) nothing, (2) a single point, (3) *a straight line*, (4) a single plane, (5) a pair of parallel planes, (6) a pair of intersecting planes, (7) *a parabolic cylinder*, (8) *an elliptic cylinder*, (9) *a hyperbolic cylinder*, (10) *an elliptical cone*, (11) *an elliptical paraboloid*, (12) *a hyperbolic paraboloid*, (13) *an ellipsoid*, (14) *a hyperboloid of one sheet*, and (15) *a hyperboloid of two sheets*.

Serret-Frenet Formulae

Space Curves

Let **r**(t) = (x(t), y(t), z(t))¢ , a £ t £ b, denote a *space curve*. Consider two points P and Q on the curve associated with the values t and t+D t. The vector obtained by joining the point P to the point Q is by D **r** = **r**(t+D t)-**r**(t). The *arc-length* s(t) is the length of the curve between the points **r**(a) and **r**(t) and is given by

s(t) = ò _{(a,t)} |d**r**(t)/dt|dt,

and the rate of change of arc-length with respect to the parameter t is ds/dt = |d**r**(t)/dt|. The rate of change of the position vector **r**(t) of the point P on the curve with respect to t is the vector d**r**(t)/dt = limD _{t® 0} D **r**/D t, which gives the direction of the *tangent* to the curve at P.

The Unit Tangent and the Principal Normal

The rate of change of **r**(t) with respect to the arc-length parameter is: **r¢ = **d**r**/ds = (d**r**/dt)/(ds/dt) = (d**r**(t)/dt)/|d**r**(t)/dt| = **u**. Here, we follow the convention of using the prime to denote the derivatives with respect to the arc-length parameter. Since it is of unit magnitude, it is called the *unit tangent vector* on the curve at P. The plane in which the tangent locally moves is the plane containing the unit tangent **u** and its rate of change d**u**/dt.

Differentiating the relation **u**^{2} º |**u**|^{2} = **u× u** = 1, we find **u× u¢ **= 0, i.e., the rate of change of the unit tangent is in a direction orthogonal to that of the unit tangent (as is the case with all unit vectors). This gives the direction of the *principal normal* to the curve at P with the *unit principal normal* **p** given by

**p** = **u¢ **/|**u¢ **| = (d**u**/dt)/|d**u**/dt|.

The quantity |**u****¢ **| is called the *curvature* of the curve at P and is denoted by the Greek letter kappa k , so that

**u¢ **= k **p**,

so that the unit tangent changes in the direction of the principal normal with the curvature as the proportionality constant. Note that for a linear curve the curvature k equals zero and the direction of the principal normal **p** remains undefined.

The Osculating, Normal and Rectifying Planes and the Unit Binormal

The plane containing the unit tangent **u** and and the principal normal **p** is called the *osculating plane* and the unit vector **u****´ p** the unit binormal **b** so that: **b** = **u´ p**. The plane containing principal normal **p** and the binormal **b** is called the *normal plane*, and the one containing the unit tangent **u** and the binormal **b** is the *rectifying plane*.

The rate of change of the principal normal lies in the plane of **u** and **b**, so that **p****¢ **= t **b** – k**u**, for some constants t and k. Using **p** = **b****´ u**, **p****¢ **= **b****¢ ´ u** + **b****´ u¢ **= (**u****´ p**)¢ ´ **u** + **b****´ k p** = [(**u****¢ ´ p**)´ **u** + (**u****´ p¢ **)´ **u**] - k **u** =(**u****´ p¢ **)´ **u** - k **u** = (**u****´ **[t **b**-k**u**])´ **u** - k **u** = t (-**p**)´ **u** - k **u** = t **b** - k **u**, so that k = k and we have

**p¢ **= t **b** – k **u**.

As, **b****¢ **= **u****¢ ´ p** + **u****´ p¢ **= **u****´ p¢ **= **u****´ **(t **b** – k **u**), we have

**b¢ **= -t **p**.

The quantity t is called the torsion of the curve at P and it follows that the binormal changes in the direction of the normal with the proportionality constant -t . Note that for a planar curve the torsion t equals zero.

The relations: **u¢ **= k **p**, **p¢ **= t **b** – k **u**,** b¢ **= -t **p**, are known as the *Serret-Frenet formulae*.

Note that by definition k ³ 0, and, |**r****² **| = k . As, **r****¢ **= **u**, **r****² **= k **p**, **r****¢ ¢ ¢ **= k ¢ **p** + k [t **b** - k **u**], [**r****¢ r² r¢ ¢ ¢ **] = **u****× **(k **p****´ **[k ¢ **p** + k (t **b**-k **u**)]) = **u****× **[k ^{2}(t **u** + k **b**)] = k ^{2}t , So that we have the formula: t = [**r¢ r² r¢ ¢ ¢ **]/k ^{2}.

Also, for differentiation with respect to the parameter t, d**r**/dt = **r****¢ **(ds/dt) = **u**(ds/dt), d^{2}**r**/dt^{2} = **r****² **(ds/dt)^{2} + **r****¢ **(d^{2}s/dt^{2}) = k **p**(ds/dt)^{2} + **u**(d^{2}s/dt^{2}), d^{3}**r**/dt^{3} = **r****¢ ¢ ¢ **(ds/dt)^{3} + 3**r****¢ ¢ **(ds/dt)(d^{2}s/dt^{2}) + **r****¢ **d^{3}s/dt^{3} = (k ¢ **p**+k [t **b** - k **u**]) (ds/dt)^{3} + 3k **p**(ds/dt)(d^{2}s/dt^{2}) + **u**(d^{3}s/dt^{3}), i.e., d^{3}**r**/dt^{3} = [-k ^{2}(ds/dt)^{3}+(d^{3}s/dt^{3})]**u** + [k ¢ +(ds/dt)^{3}+3k (ds/dt) (d^{2}s/dt^{2})]**p** + k t **b**.

Some inner products and a scalar triple product involving derivatives with respect to the parameter t are:

(d**r**/dt)^{2} = (ds/dt)^{2}, (d**r**/dt)× (d^{2}**r**/dt^{2}) = (ds/dt)× (d^{2}s/dt^{2}), (d^{2}**r**/dt^{2})^{2} = k ^{2}((ds/dt)^{4} + (d^{2}s/dt^{2})^{2}.

[d**r**/dt d^{2}**r**/dt^{2} d^{3}**r**/dt^{3}] = (**u**ds/dt´ [k **p**(ds/dt)^{2} + **u**(d^{2}s/dt^{2})])× d^{3}**r**/dt^{3} = k (ds/dt)^{3}**b****× **(k t **b)** = k ^{2}t (ds/dt)^{3}.

As, [(d**r**/dt)^{2}][(d^{2}**r**/dt^{2})^{2}]–[(d**r**/dt)× (d^{2}**r**/dt^{2})]^{2} = (ds/dt)^{2}[k ^{2}((ds/dt)^{4}+(d^{2}s/dt^{2})^{2}]–[(ds/dt)× (d^{2}s/dt^{2})]^{2} = k ^{2}((ds/dt)^{6},

, .

Velocity and Acceleration

If r(t) denotes the position vector of a moving point in the space at time t, and v(t), a(t) its velocity, and acceleration vectors, respectively, we have

v(t) = dr/dt = (ds/dt)u,

a(t) = dv/dt = d^{2}r/dt^{2} = (d^{2}s/dt^{2})u + k (ds/dt)^{2}p,

from which we observe that the direction of the velocity of the particle is along the tangent of its trajectory, but that its acceleration has two components (i) *the tangential accelearation* (d^{2}s/dt^{2})u, along the direction of the tangent, and, (ii) *the normal accelearation* k (ds/dt)^{2}p, along the direction of the principal normal. This second term is due to the curvature k of the trajectory. This normal acceleration is related with the terms known as *centripetal* and *Coriolis accelerations*, respectively, in a circular motion with a constant angular velocity, and, in a motion with constant speed along a radial line on a rotating disk, or along a meridian on a rotating sphere..