Book:L. Harwood Clarke/A Note Book in Pure Mathematics
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L. Harwood Clarke: A Note Book in Pure Mathematics
Published $\text {1953}$, William Heinemann Ltd.
Contents
- Foreword
- $\text {I}$. ALGEBRA
- Permutations and Combinations
- The value of ${}_n P_r$
- The number of ways of arranging $n$ things in a line
- The number of ways of seating $n$ people at a circular table
- The number of ways of threading $n$ beads on a wire
- The value of ${}_n C_r$
- The number of ways of arranging $n$ things in line if $p$ are alike of one kind and $q$ are alike of another kind
- The number of ways of dividing $\paren {p + q + r}$ things into three unequal groups, the first to contain $p$ things, the second $q$ things and the third $r$ things
- The number of ways of dividing $3 p$ things into three equal groups each containing $p$ things
- The number of selections from $n$ things if any number may be taken
- The number of selections from $n$ different things, $p$ similar things of one kind and $q$ similar things of another kind, if any number may be taken
- ${}^n C_r = {}_n C_{n - r}$
- ${}^n C_r + {}_n C_{r + 1} = {}_{n + 1} C_{r + 1}$
- Permutations and Combinations
- The Method of Induction
- The Binomial Theorem
- Relations between coefficients
- Negative and fractional indices
- Approximations
- The greatest term
- The Binomial Theorem
- Partial Fractions
- Summation of Series
- The sum of the squares of the first $n$ integers
- The sum of the cubes of the first $n$ integers
- Power series
- Summation of Series
- The Exponential and Logarithmic Series
- Applications to summing series
- The Exponential and Logarithmic Series
- The Remainder Theorem
- Roots of Equations
- The quadratic function
- The condition that $\paren {a x^2 + b x + c}$ should be positive for all values of $x$
- The condition for a common root
- The condition for a repeated root
- Relations between roots
- To form the equation whose roots are symmetrical functions of each of the roots of a given equation
- To form the equation whose roots are any symmetrical functions of the roots of a given equation
- The sum of the powers of the roots of a given equation
- Finding numerical roots
- The function $\dfrac {a x^2 + b x + c} {A x^2 + B x + C}$ when $x$ is real
- Roots of Equations
- $\text {II}$. CALCULUS
- Differentiation
- Differential coefficient
- Standard differential coefficients
- Double function
- Product
- Quotient
- Variable index
- Inverse ratios
- Differentiation
- Maximum, Minimum and Point of Inflection
- Velocity and Acceleration
- Tangent and Normal
- Parametric Equations
- Differentiation $n$ Times
- Leibnitz' Theorem
- Maclaurin's Theorem
- Taylor's Theorem
- Integration
- Algebraic integration
- Powers of cos and sine
- Useful substitutions
- Integration by parts
- Integration
- Definite Integrals
- Integral of $1 / x$
- Area, Volume and Centre of Gravity
- Arcs
- Polar Coordinates
- Moments of Inertia
- Reduction Formulae
- Hyperbolic Functions
- Differential Equations
- $\text {III}$. ANALYTICAL GEOMETRY
- The Straight Line
- Point dividing $AB$ in a given ratio
- Centre of gravity of a triangle
- The area of a triangle
- Forms for the equation of a straight line
- The angle between two lines
- The length of the perpendicular
- The equations of the angle bisectors
- Lines through the intersection of two given lines
- The Straight Line
- Pairs of Straight Lines
- Their equation
- The angle between the pair
- The equation of the angle bisectors
- The condition for a pair
- Pairs of Straight Lines
- The Circle
- Its equation
- The tangent of gradient $m$
- The tangent at $\tuple {x', y'}$
- The polar of $\tuple {x', y'}$
- Orthogonal circles
- The length of a tangent
- Coaxal circles and radical axes
- The Circle
- The Parabola
- Its equation
- The tangent of gradient $m$
- The normal
- The equation of a chord
- The tangent at $\tuple {x', y'}$
- The locus of the foot of the perpendicular from the focus to a tangent
- The locus of the intersection of perpendicular tangents
- The polar of $\tuple {x', y'}$
- The feet of the normals from a point
- The locus of the mid points of parallel chords
- The chord of mid point $\tuple {x', y'}$
- The Parabola
- The Ellipse and Hyperbola
- Their equations
- The tangent of gradient $m$
- The auxiliary circle
- The director circle
- The tangent at $\tuple {x', y'}$
- The normal at $\tuple {x', y'}$
- The polar of $\tuple {x', y'}$
- The pole of $l x + m y + n = 0$
- The locus of the mid points of parallel chords
- Conjugate diameters
- The chord of mid point $\tuple {x', y'}$
- The Ellipse and Hyperbola
- The Ellipse
- The eccentric angle
- The line joining $\alpha$ and $\beta$
- Eccentric angles at the ends of conjugate diameters
- The Ellipse
- The Hyperbola
- The asymptotes
- Parametric representation
- The Hyperbola
- The Rectangular Hyperbola
- Its equation
- The tangent at $\tuple {x', y'}$
- The polar of $\tuple {x', y'}$
- The chord joining $m$ and $l$
- The locus of the mid points of parallel chords
- The chord of mid point $\tuple {x', y'}$
- The equation of a hyperbola with given asymptotes
- The Rectangular Hyperbola
- The General Conic
- The conditions for: a pair of straight lines, a circle, a parabola, an ellipse, a hyperbola, a rectangular hyperbola
- A conic through the intersections of two given conics
- The polar of $\tuple {x', y'}$
- The General Conic
- $\text {IV}$. PURE GEOMETRY
- Plane Geometry
- The incentre
- The circumcentre
- The orthcentre
- The centre of gravity
- The centre of similitude
- Apollonius' Circle
- Ptolemy's Theorem
- The Euler Line
- The Nine-Point Circle
- Ceva's Theorem
- Menelaus' Theorem
- The Simson Line
- The radical axis and coaxal circles
- Plane Geometry
- Solid Geometry
- The plane
- Skew lines
- Generators
- The angle between a line and a plane
- The angle between two planes
- If a line is perpendicular to each of two intersecting lines, it is perpendicular to their plane
- The tetrahedron
- Its circumscribing parallelepiped
- Its circumscribing sphere
- The common perpendicular to two skew lines
- The intersection of a sphere and a plane
- The intersection of two spheres
- Solid Geometry
- Orthogonal Projection
- Lines
- Centre of Gravity
- Areas
- The circle
- Geometrical properties of the ellipse
- Orthogonal Projection
- $\text {V}$. TRIGONOMETRY
- Definitions of the Ratios
- Special Angles
- Complementary Angles
- Angles Larger than $90 \degrees$
- The Cosine Formula and the Sine Formula
- The Addition Formulae
- Tangents of Sum and Difference
- The Product Formulae
- Factorization of Sums of Sines and Cosines
- The Auxiliary Angle
- Solution of Equations
- Solution of Triangles
- Inverse Ratios
- Half Angle Formulae
- Area of the Triangle
- The Median and the Centre of Gravity
- The Orthocentre
- The Angle Bisector
- The Pedal Triangle
- The Circumcircle
- The Incircle
- The Ex-circles
- ANSWERS
Click here for errata
Source work progress
- From Next:
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous): $\text V$. Trigonometry: The ex-circles
- From Next:
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous): $\text V$. Trigonometry: Area of the triangle
- From Next:
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Solution of triangles
- From Next:
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {IV}$. Pure Geometry: Plane Geometry: The Centre of Similitude
- From Next:
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {III}$. Analytical Geometry: The Straight Line: The angle between two lines: Example $\text{(ii)}$
- From Next:
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XV}$: $3$
- From Next:
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Maximum, Minimum and Point of Inflection
- From the start:
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: The Binomial Theorem: Exercises $\text {III}$: $1 \ \text {(d)}$