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Further Pure 3 (Differentiation (\(\frac{d}{dx}\cosh(x)=\sinh(x)\),…
Further Pure 3
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Matrices
Symmetric Matrices
\[M^{T}=M\]
- Real Hermitian Matrices are symmetric and all symmetric matrices are Hermitian
- Can be diagonalised to form a diagonal matrix
Hermitian Matrices
\[M^\dagger=M\]
- Are their own conjugate transposes
- For an \(N\times N\) Hermitian matrix there are \(N\) distinct orthogonal Eigenvectors
Diagonalising a Square Symmetric Matrix \(A\)
- Find the set of orthogonal normalised Eigenvectors of the matrix \(A\)
- Form an orthogonal matrix, \(P\)
\[D=P^{-1}AP=P^{T}AP\]
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Orthogonal Matrices
\[MM^{T}=I\]\[\therefore\;M^{T}=M^{-1}\]
- Have columns composed of orthogonal unit vectors
- Necessarily invertable
Diagonal Matrix
- All non-zero entries lie along the main diagonal
- Symmetric
- Determinant equals the product of the Eigenvalues
Determinant
- The area scale factor of a matrix
\(\mathrm{Det}{AB}=\mathrm{Det}(A)\,\mathrm{Det}(B)\)
Inverse of a matrix
\[\frac{1}{\mathrm{Det}(M)}C^{T}\]
Where \(C\) is the matrix of cofactors (minors with appropriate sign change using diamond of negatives)
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Vectors
Vector Product
\({\bf a}\times{\bf b}=|{\bf a}||{\bf b}|\sin{\theta}\,{\bf \hat{n}}\)
Where \(\bf\hat{n}\) is normal to both \(\bf a\) and \(\bf b\)
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Lines
Vector Equation of a Line
\({\bf r}={\bf a}+\lambda{\bf b}\)
Where \(\bf a\) is a point on the line and \(\bf b\) is the direction vector
Cross Product Form
\(({\bf r} - {\bf a})\times{\bf b}=0\)
(This leads to the vector equation as \({\bf r}-{\bf a}\) must be parallel to \(\bf b\)
Cartesian Form
\[\frac{x-x_{1}}{l}=\frac{y-y_{1}}{m}=\frac{z-z_{1}}{n}\]
Where \(\begin{pmatrix}x_{1} \\ y_{1} \\ z_{1}\end{pmatrix}\) is the position vector of a point on the line, and \(\begin{pmatrix}l \\ m \\ n\end{pmatrix}\) is the direction vector
Planes
Dot Product Form
\({\bf r}\cdot{\bf n}={\bf a}\cdot{\bf n}=p\)
Where \(\bf a\) is the position vector of a point in the plane and \(\bf n\) is a vector normal to the plane
Vector Equation of a Plane
\({\bf r}={\bf a}+\lambda{\bf b}+\mu{\bf c}\)
Where \(\bf a\) is the position vector of a point in the plane, and \(\bf b\) and \(\bf c\) are two non-parallel vectors that lie parallel to the plane
Cartesian Form
\[ax+by+cz+d=0\]
Where \(\begin{pmatrix} a\\b\\c \end{pmatrix}\) is a vector normal to the plane
Shortest Distance Between Two Skew Lines \({\bf r}={\bf a}+\lambda{\bf b}\) and \({\bf r}={\bf c}+\lambda{\bf d}\)
\[d=\frac{({\bf a}-{\bf c})\cdot({\bf b}\times {\bf d})}{{\bf b}\times {\bf d}}\]
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Hyperbolic Functions
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Osborne's Rule
The hyperbolic functions follow the same identities as their corresponding trigonometric, but when there is a product of \(\sinh\) you change the sign of that term
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Coordinate Systems
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Hyperbola
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\(x=\pm\,a\cosh(t)\)
\(y=b\sinh(t)\)
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Intersects the x-axis at \((\pm\,a, 0)\)
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Ellipse
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Intersects the x-axis at \(\pm\,a\)
Intersects the y-axis at \(\pm\,b\)
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Integration
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Use integration by parts to try and find an expression for an integral in terms of a simpler integral, in a recursive fashion
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Surface Area of Revolution
\[2\pi\int y\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx = 2\pi\int y\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt\]
Arc Length
\[\int \sqrt{1+\left(\frac{dy}{dx}\right)^2}dx = \int \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt\]