• The Catenary

    Suppose a length \[L\] of chain is hung from an uneven ceiling. The ends of the chain are fixed to two points some distance apart. What will be the shape of the chain? Let the chain lie in the vertical \[xy\] plane. An small length \[ds\] of chain of...

    https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/5425-the-catenary.html

  • Time and Distance To Interception

    A boat starts from A \[5 \mathbf{i} + 12 \mathbf{j}\] with a speed of 12 km/h on a bearing of 70 degrees. Another boat starts from the origin with a speed of 15 km/h to intercept the first ship. How long will it be before interception takes place, and...

    https://astarmathsandphysics.com/a-level-maths-notes/m4/5078-time-and-distance-to-interception.html

  • Lamina Suspended at Angle to Horizontal

    In general for a system to be in equilibrium, the forces must balance in each direction, and the moments about any point must sum to zero. When a lamina is suspended by two inelastic strings, we can find the tensions in the strings by resolving...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3640-lamina-suspended-at-angle-to-horizontal.html

  • Multi Stage Rockets

    It is very expensive to get a mass into orbit. A lot of the mass of a rocket is useless in space. For this reason rockets are not usually sent into space as complete units, but are divided into stages, with each stage acting as a store of fuel and a...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3646-multi-stage-rockets.html

  • Modelling Average Unit Maintenance Cost

    A certain factory uses crates to pack bottles of soda pop. The condition of crates falls into one of four categories - good, fair, poor, broken. If a crate is broken it must be repaired, which costs £2.50, takes a week, and costs £1.85 in lost...

    https://astarmathsandphysics.com/university-maths-notes/matrices-and-linear-algebra/5093-modelling-average-unit-maintenance-cost.html

  • Modelling Cashflow and Bad Debt

    A trader extends credit to his customers. The credit extended falls into four categories - paid, bad debt, owing - if within a thirty day period - or overdue - if outside the period, up to thirty more days. Any money owing after this period becomes a...

    https://astarmathsandphysics.com/university-maths-notes/matrices-and-linear-algebra/5094-modelling-cashflow-and-bad-debt.html

  • Properties of Euler's Totoent Function

    Euler's totient function is defined as ie the number of positive integers less than or equal to relatively prime to The function has the following properties: a) b) (1) where c) if d) divides implies divides e) is even for Also if has distinct odd...

    https://astarmathsandphysics.com/university-maths-notes/number-theory/1975-properties-of-euler-s-totoent-function.html

  • The Euclidean Algorithm

    The greatest common divisor of two numbers and may be found from the prime – power factorizations of and are known. Considerable computing power may be needed to find the prime power factorizations however and there is a procedure – Euclid's algorithm...

    https://astarmathsandphysics.com/university-maths-notes/number-theory/1983-the-euclidean-algorithm.html

  • Proof That a Linear Operatior on a Vector Space to the Set of Real Numbers is the Inner Product

    Let \[\mathbb{R}^n\] be the vector space of real n-tuples of the form \[(x_1,x_2,...,x_n)\] . Let the inner product \[\phi\] on this vector space satisfy \[(x_1,x_2,...,x_n) \cdot (y_1,y_2,...,y_n)=x_1y_1+x_2y_2+...+x_ny_n\] The inner product defined...

    https://astarmathsandphysics.com/university-maths-notes/matrices-and-linear-algebra/5036-proof-that-a-linear-operatior-on-a-vector-space-to-the-set-of-real-numbers-is-the-inner-product.html

  • Bodies on Slopes Toppling

    A body will topple when the vertical through the centre of gravity lies just outside the base. A particle on a slope is typically more susceptible to toppling because the vertical is closer to the point where it would lie just outside the base. The...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3630-bodies-on-slopes-toppling.html

  • Maximum Extension of Elastic String Attached to Particle on Rough Plane

    When a particle is attached to a an elastic string on a rough plane, it will come to rest at a point on the slope where whatever gravitational potential energy the particle started with has been turned into elastic potential energy in the string of...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3642-maximum-extension-of-elastic-string-attached-to-particle-on-rough-plane.html

  • Proof That if d Divides p-1, the Congruence x^d-1 =0 mod p has Exactly d Solutions

    Theorem If \[p\] is a prime and \[d\] is a divisor of \[p-1\] then the congruence \[x^d-1 \equiv 0 \; (mod \; p)\] has exactly \[d\] solutions. Proof Let \[p-1=dr\] for some integer \[r\] . If \[r-1\] then \[d=p-1\] and the result follows from t...

    https://astarmathsandphysics.com/university-maths-notes/number-theory/5151-proof-that-if-d-divides-p-1-the-congruence-x-d-1-0-mod-p-has-exactly-d-solutions.html

  • Equation for Legendre Symbol

    Theorem If \[p\] is an odd prime then \[(a/p)=(-1)^{\alpha (a, \; p)}\] where \[\alpha (a, \; p)= \sum_{k=1}^{(p-1)/2} int (\frac{ka}{p} )\] Proof Let \[S= \{a, \; 2a, \; 3a,..., \; \frac{p-1}{2} \}\] . Replace each element of \[S\] by its least residue...

    https://astarmathsandphysics.com/university-maths-notes/number-theory/5172-equation-for-legendre-symbol.html

  • Uniqueness Infinite Continued Fraction

    Theorem (Uniqueness of the Infinite Continued Fraction) Given any irrational number \[x\] determined from \[x\] by the continued fraction algorithm converges to \[x\] and no other finite continued fraction converges to \[x\] . Proof Using the continued...

    https://astarmathsandphysics.com/university-maths-notes/number-theory/5191-uniqueness-infinite-continued-fraction.html

  • Approximate Square Roots

    We can find quite an accurate estimate of a square root using simple algebra. The square root of \[1300\] is approximately 36 - \[36^2=1296\] , so the square root is just a little bit more. We can find a better estimate by writing \[\sqrt{1300}=36+x\]...

    https://astarmathsandphysics.com/ib-maths-notes/sequences-and-series/5030-approximate-square-roots.html

  • Linear Functional - Definition and Examples

    A mapping \[\phi\] from a vector space \[V\] to a field \[F\] (often the set of real numbers ( \[\mathbb{R}\] ) is called a linear functional if for every \[\mathbf{v}_1, \; \mathbb{R} \in V\] , \[\phi(\mathbb{v}_1 + \mathbb{v}_2)= \phi (\mathbb{v}_1)_...

    https://astarmathsandphysics.com/university-maths-notes/matrices-and-linear-algebra/5041-linear-functional-definition-and-examples.html

  • Elastic Strings

    Elastic strings are similar to springs. Both strings and springs obey Hooke's Law when being extended, but when being compressed, there is no tension in the string while the spring experiences a force tending to increase the length. (Springs obey...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3639-elastic-strings.html

  • Maximum Angular Speed for Horizontal Circular Motion of a Particle on an Elastic String

    A particle moving in a horizontal circle, suspended from the ceiling by an elastic string of natural length and modulus of elasticity must move with an angular velocity less than some minimum Applying horizontally gives (1) so (1) can be written hence...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3641-maximum-angular-speed-for-horizontal-circular-motion-of-a-particle-on-an-elastic-string.html

  • Simple Harmonic Motion for a Particle Suspended on an Elastic String

    Suppose we suspend a particle of mass m from an elastic spring. The tension in the string is The particle will be in equilibrium when the tension is equal to the weight. We have where x-e labels the equilibrium extension. If the particle is pulled down...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3653-simple-harmonic-motion-for-a-particle-suspended-on-an-elastic-string.html

  • Associating Multilinear With Alternating Forms

    Let \[V\] be a vector space over a field \[F\] , let \[r\] be a positive integer.and let \[L\] be a multilinear function (linear in each argument) \[L:V^r \rightarrow F\] . \[L\] is an alternating form if \[L=0\] whenever two arguments are the same, and...

    https://astarmathsandphysics.com/university-maths-notes/matrices-and-linear-algebra/5055-associating-multilinear-with-alternating-forms.html

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