ENGR 2422 Engineering Mathematics 2
Brief Notes on Chapter 1

1.1 Lines and Planes
1.2 Polar Coordinates and
velocity and acceleration, [not in 2004 Winter - covered in ENGR 1405]
1.3 Area, Arc Length and Curvature
1.4 Classification of Conic Sections (simplest cases only)
1.5 Classification of Quadric Surfaces (simplest cases only)
1.6 The Equation and Area of a Surface of Revolution
1.7 Hyperbolic functions and their comparison to trigonometric functions
1.8 Integration by Parts
1.9 Leibnitz Differentiation of an Integral

1.1 Lines and Planes

A plane is a two dimensional object.
The orientation of a plane in real ³ is determined by any non-zero normal vector n.
Knowledge of the position vector a to any point (x_o, y_o, z_o) on the plane Capital Pi then fixes its position.
The general point (x, y, z) with position vector r is also on the plane if and only if

r•n = a•n

(which is the vector equation of the plane).

If the Cartesian components of n are (A, B, C), then the Cartesian equation of the plane is

Ax + By + Cz + D = 0

(where D = –a•n = –(Ax_o + By_o + Cz_o) ).
Given three non-collinear points A, B, C in a plane, more vector equations for the plane are:
AR dot AB cross AC = 0
where R is the general point (x, y, z);
and the two-parameter vector form
r = a + s u + t v ,
where s and t are any real numbers,
a is any one of vectors OA, OB and OC
and u and v are any two of vectors AB, BC and CA .

A line is a one dimensional object.
The orientation of a line is determined by any non-zero tangent (or direction) vector v.
Knowledge of the position vector a to any point (x_o, y_o, z_o) on the line then fixes its position.
The general point (x, y, z) with position vector r is also on the line if and only if

r = a + t v ,

the vector parametric form, where t is any real number.

The symmetric Cartesian equation of the line is

(x-xo)/v1 = (y-yo)/v2 = (z-zo)/v3

(except when one or more of v₁, v₂, v₃ are zero),
where v₁, v₂, v₃ are the Cartesian components of the tangent vector v.

The angle theta between any two lines is also the angle between their direction vectors v₁, v₂:

cos theta = (vector v1 dot vector v2) / (|v1| |v2|)

1.2 Polar Coordinates

(not in 2004 Winter – covered in ENGR 1405)

If (r, theta ) is a pair of polar coordinates for a point not at the pole, then so are
(r, theta + 2n) and (–r, theta + (2n + 1)).
(0, theta ) is at the pole for any value of theta .

Conversion between polar and Cartesian:

If x < 0 and y > 0 (2^nd quadrant), then
theta = tan^–1(y / x) +
If x < 0 and y < 0 (3^rd quadrant), then
theta = tan^–1(y / x) –
If x > 0 then
theta = tan^–1(y / x)

triangle, showing relationship between
Cartesian (x, y) and polar (r, theta)

A point is on a curve r = f ( theta ) if and only if at least one member of the set
{ (r, theta + 2n) and (–r, theta + (2n + 1)) }
satisfies the equation r = f ( theta ).

The slope at any point on a curve, whose equation r = f ( theta ) is expressed in the polar parametric form (x, y) = (f ( theta ) cos theta , f ( theta ) sin theta ), is

dy/dx = (f'(t) sin t + f(t) cos t) /
(f'(t) cos t - f(t) sin t)

If r -> 0 but dr/dt -> not 0 as t -> to then the line theta = theta _o is a tangent to the curve at the pole.
To sketch a polar curve, divide the range of values of theta into intervals at whose endpoints one or more of the following is true:
r = 0
r becomes undefined
or dr/d theta = 0 (or becomes undefined)
Then follow the behaviour of r as theta increases inside each interval.

The arc length along the curve r = f ( theta ) from theta = alpha to theta = beta is

$L = Integ{ sqrt(r^2 + (dr/dt)^2) } dt$

[Be careful to take the positive square root throughout the range of integration.]

The area swept out by the radius vector is

$A = (1/2) * Integ{ r^2 } dt$

Radial and Transverse Components of Velocity and Acceleration

(not in 2004 Winter – covered in ENGR 1405)

Let the unit vectors at a point in real ² in the radial and transverse directions be rHat and thetaHat respectively, so that the displacement vector of the point is vector r = scalar r * rHat . Also use the “over-dot” notation for differentiation with respect to the parameter t: vDot = dv/dt .
d(rHat)/dt = thetaDot thetaHat and
d(thetaHat)/dt = -thetaDot rHat
If t represents time, then the velocity is vector v = rDot rHat + r thetaDot thetaHat .
The radial component of velocity is rDot .
The transverse component of velocity is .
The radial component of acceleration is rDotDot - r thetaDot^2 .
The transverse component of acceleration is 2 rDot thetaDot + r thetaDotDot .

For future reference in other courses (but not required in ENGR 2422),
in spherical polar coordinates (r, theta , phi ) in real ³:

vector v = rDot rHat + r(thetaDot thetaHat + sin theta phiDot phiHat)
vector a = [lengthy expression]

1.3 Area, Arc Length and Curvature

The area A of a region in real ² bounded by the lines y = 0, x = a, x = b, (a < x < b) and the curve y = f (x) (with f (x) > 0) is

A = Integ { f(x) } dx

If the position on the curve is expressed parametrically as (x(t), y(t)), then this formula becomes

A = Integ { y * dx/dt } dx

where x(t_a) = a and x(t_b) = b.

When the position of a point on a curve in real ³ is given as a vector function of one scalar parameter, r(t), then the tangent vector is

T = dr/dt ;
unit tangent = T / |T|

v = T

v = T = | T |

The distance travelled along the curve is the arc length s .

$ds/dt = sqrt{(dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2}$

so that the element of arc length is

$ds = sqrt{(dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2} dt$

If t is time, then v = ds / dt is the speed.

The length L along a curve from a point where the parameter value is t₀ to a point where the parameter value is t₁ is

L = Integ | d(vector r)/dt) | dt

The principal normal vector N is the rate of change of the unit tangent vector with respect to the distance travelled along the curve:

N = d(T^)/ds = d(T^)/dt / | dr/dt |

The magnitude of the principal normal vector is the curvature:

kappa = | N | = | d(T^)/dt | / | dr/dt |
and N = kappa N^

In terms of the displacement vector r(t) and denoting differentiation with respect to the parameter t by the overdot notation, another formula for the curvature is

kappa = | rDot cross rDotDot | / | rDot |^3

The radius of curvature is the reciprocal of the curvature:

rho = 1 / kappa

At any point on a curve, an orthonormal basis for real ³ can be constructed, using the unit tangent, principal unit normal and unit binormal vectors, with B^ = T^ cross N^ .

1.4 Classification of Conic Sections (simplest cases only)

The eccentricity e is a parameter related to the slope of the plane relative to the cone.

e Type Standard
equation Location, other details

e = 0 circle x² + y² = r² centre O, radius r

0 < e < 1 ellipse x^2/a^2 + y^2/b^2 = 1 centre O, vertices (±a, 0)
semi-major axis a,
semi-minor axis b = a square root (1 – e²),
foci at (±ae, 0)

e = 1 parabola y² = 4ax vertex at O, focus at (a, 0)

e > 1 hyperbola x^2/a^2 - y^2/b^2 = 1 centre O, vertices (±a, 0),
asymptotes y = ± bx / a,
foci at (±ae, 0)

e = 2 rectangular
hyperbola xy = k centre O, axes are asymptotes

Degenerate conic sections (the plane passes through the apex of the cone)

0 < e < 1 point x^2/a^2 + y^2/b^2 = 0 at O

e > 1 line pair x^2/a^2 - y^2/b^2 = 0 through O, y = ± bx / a

1.5 Classification of Quadric Surfaces (simplest cases only)

x^2/a^2 + y^2/b^2 + z^2/c^2 = 1 Ellipsoid (Axis lengths a , b , c )

x^2/a^2 + y^2/b^2 - z^2/c^2 = 1 Hyperboloid of One Sheet (Ellipse axis lengths a , b;
aligned along the z axis)

x^2/a^2 - y^2/b^2 - z^2/c^2 = 1 Hyperboloid of Two Sheets (Ellipse axis lengths b , c;
aligned along the x axis)

z/c = x^2/a^2 + y^2/b^2 Elliptic Paraboloid (Ellipse axis lengths a , b;
aligned along the z axis)

z/c = x^2/a^2 - y^2/b^2 Hyperbolic Paraboloid (Hyperbola axis length a;
aligned along the z axis)

	Ellipsoid	(Axis lengths a , b , c )
	Hyperboloid of One Sheet	(Ellipse axis lengths a , b; aligned along the z axis)
	Hyperboloid of Two Sheets	(Ellipse axis lengths b , c; aligned along the x axis)
	Elliptic Paraboloid	(Ellipse axis lengths a , b; aligned along the z axis)
	Hyperbolic Paraboloid	(Hyperbola axis length a; aligned along the z axis)

Degenerate cases:

x^2/a^2 + y^2/b^2 + z^2/c^2 = 0 Point

x^2/a^2 + y^2/b^2 - z^2/c^2 = 0 Elliptic Cone

Nothing

x^2/a^2 + y^2/b^2 = 1 Elliptic Cylinder

x^2/a^2 + y^2/b^2 = 0 Line

x^2/a^2 + y^2/b^2 = -1 Nothing

x^2/a^2 - y^2/b^2 = 1 Hyperbolic Cylinder

x^2/a^2 - y^2/b^2 = 0 Intersecting Plane Pair

y/b = x^2/a^2 Parabolic Cylinder

x^2/a^2 = 1 Parallel Plane Pair

x^2/a^2 = 0 Plane

x^2/a^2 = -1 Nothing

xz/a = y^2/b^2 Parabolic Cone

1.6 The Equation and Area of a Surface of Revolution

If the curve y = f (x) is rotated once about the line y = c, then the equation of the surface in real ³ generated by this revolution is

(y – c)² + z² = (f (x) – c)²

The curved surface area (excluding the circular cross-sections at both ends) of this surface of revolution, between x = a and x = b, is

$As = 2*pi* Integ{ |f - c| sqrt(1 + (dy/dx)^2) } dx$

1.7 Hyperbolic functions and their
comparison to trigonometric functions

Trigonometric identities Hyperbolic function identities

exp(jt) = cos t + j sin t e^x = cosh x + sinh x

cos x = cosh jx cosh x = cos jx

sin x = -j sinh jx sinh x = -j sin jx

tan x = sin x / cos x tanh x = sinh x / cosh x

sec x = 1 / cos x sech x = 1 / cosh x

csc x = 1 / sin x csch x = 1 / sinh x

cot x = 1 / tan x coth x = 1 / tanh x

cos (–x) = + cos x cosh (–x) = + cosh x

sin (–x) = – sin x sinh (–x) = – sinh x

tan (–x) = – tan x tanh (–x) = – tanh x

cos²x + sin²x = 1 cosh²x – sinh²x = 1

sec²x = 1 + tan²x sech²x = 1 – tanh²x

csc²x = 1 + cot²x csch²x = coth²x – 1

(d/dx) cos x = - sin x (d/dx) cosh x = + sinh x

(d/dx) sin x = cos x (d/dx) sinh x = cosh x

(d/dx) tan x = sec^2 x (d/dx) tanh x = sech^2 x

(d/dx) sec x = sec x tan x (d/dx) sech x = - sech x tanh x

(d/dx) csc x = - csc x cot x (d/dx) csch x = - csch x coth x

(d/dx) cot x = - csc^2 x (d/dx) coth x = - csch^2 x

cos (A+B) = cos A cos B – sin A sin B cosh (A+B) = cosh A cosh B + sinh A sinh B

cos 2x = cos²x – sin²x
= 2 cos²x – 1 = 1 – 2 sin²x cosh 2x = cosh²x + sinh²x
= 2 cosh²x – 1 = 1 + 2 sinh²x

sin (A+B) = sin A cos B + cos A sin B sinh (A+B) = sinh A cosh B + cosh A sinh B

sin 2x = 2 sin x cos x sinh 2x = 2 sinh x cosh x

Trigonometric identities	Hyperbolic function identities
	e^x = cosh x + sinh x


tan x = sin x / cos x	tanh x = sinh x / cosh x
sec x = 1 / cos x	sech x = 1 / cosh x
csc x = 1 / sin x	csch x = 1 / sinh x
cot x = 1 / tan x	coth x = 1 / tanh x
cos (–x) = + cos x	cosh (–x) = + cosh x
sin (–x) = – sin x	sinh (–x) = – sinh x
tan (–x) = – tan x	tanh (–x) = – tanh x
cos²x + sin²x = 1	cosh²x – sinh²x = 1
sec²x = 1 + tan²x	sech²x = 1 – tanh²x
csc²x = 1 + cot²x	csch²x = coth²x – 1






cos (A+B) = cos A cos B – sin A sin B	cosh (A+B) = cosh A cosh B + sinh A sinh B
cos 2x = cos²x – sin²x = 2 cos²x – 1 = 1 – 2 sin²x	cosh 2x = cosh²x + sinh²x = 2 cosh²x – 1 = 1 + 2 sinh²x
sin (A+B) = sin A cos B + cos A sin B	sinh (A+B) = sinh A cosh B + cosh A sinh B
sin 2x = 2 sin x cos x	sinh 2x = 2 sinh x cosh x

Also:
$Integ { du / sqrt(a^2 + u^2) } = Arcsinh(u/a) + C$ $Integ { du / (a^2 - u^2) } = (1/a)*Arctanh(u/a) + C$
$Integ { sqrt(a^2 + u^2) du } = u/2 * sqrt(a^2 + u^2) + a^2/2 * Arcsinh(u/a) + C$

1.8 Integration by Parts

Integ(u v') = [uv] - Integ(u' v)
Tabular shortcut for Integ(f g)

[table for shortcut method
of integration by parts]

See the examples done in class and on problem sets.

Some forms that can be obtained from integration by parts:

Integ(ln u) = u(ln u - 1) + C
Integ(exp(au)*sin(bu)) = ...
Integ(exp(au)*cos(bu)) = ...
Integ(sin^m(u)*cos^n(u)) = ...

1.9 Leibnitz Differentiation of an Integral

$If I = Int_f^g {h} dt, then$
$dI/dx = h*g' - h*f' + Int_f^g {dh/dx} dt$

Special cases:
$(d/dx)Int_k^x {h} dt = h$
$(d/dx)Int_a^b {h} dt = Int_a^b {dh/dx} dt$

END OF CHAPTER 1

[Index of Brief Notes]

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Created 2004 01 06 and last modified 2004 01 07 by Dr. G.H. George

e	Type	Standard equation	Location, other details
e = 0	circle	x² + y² = r²	centre O, radius r
0 < e < 1	ellipse		centre O, vertices (±a, 0) semi-major axis a, semi-minor axis b = a (1 – e²), foci at (±ae, 0)
e = 1	parabola	y² = 4ax	vertex at O, focus at (a, 0)
e > 1	hyperbola		centre O, vertices (±a, 0), asymptotes y = ± bx / a, foci at (±ae, 0)
e = 2	rectangular hyperbola	xy = k	centre O, axes are asymptotes
Degenerate conic sections (the plane passes through the apex of the cone)
0 < e < 1	point		at O
e > 1	line pair		through O, y = ± bx / a

ENGR 2422 Engineering Mathematics 2 Brief Notes on Chapter 1