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Ceres Solver
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1.13
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Why?
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On Derivatives
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Spivak Notation
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Analytic Derivatives
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Numeric derivatives
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On Derivatives »
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Spivak Notation
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Spivak NotationΒΆ
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To preserve our collective sanities, we will use Spivak’s notation for
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derivatives. It is a functional notation that makes reading and
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reasoning about expressions involving derivatives simple.
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For a univariate function \(f\), \(f(a)\) denotes its value at
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\(a\). \(Df\) denotes its first derivative, and
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\(Df(a)\) is the derivative evaluated at \(a\), i.e
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\[Df(a) = \left . \frac{d}{dx} f(x) \right |_{x = a}\]
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\(D^kf\) denotes the \(k^{\text{th}}\) derivative of \(f\).
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For a bi-variate function \(g(x,y)\). \(D_1g\) and
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\(D_2g\) denote the partial derivatives of \(g\) w.r.t the
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first and second variable respectively. In the classical notation this
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is equivalent to saying:
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\[D_1 g = \frac{\partial}{\partial x}g(x,y) \text{ and } D_2 g = \frac{\partial}{\partial y}g(x,y).\]
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\(Dg\) denotes the Jacobian of g, i.e.,
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\[Dg = \begin{bmatrix} D_1g & D_2g \end{bmatrix}\]
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More generally for a multivariate function \(g:\mathbb{R}^n
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\longrightarrow \mathbb{R}^m\), \(Dg\) denotes the \(m\times
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n\) Jacobian matrix. \(D_i g\) is the partial derivative of
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\(g\) w.r.t the \(i^{\text{th}}\) coordinate and the
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\(i^{\text{th}}\) column of \(Dg\).
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Finally, \(D^2_1g\) and \(D_1D_2g\) have the obvious meaning
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as higher order partial derivatives.
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For more see Michael Spivak’s book Calculus on Manifolds
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or a brief discussion of the merits of this notation by
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Mitchell N. Charity.
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