Ex 3.3: Hill Climbing, a

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Functional Differential Geometry

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The topography of a region on the Earth can be specified by a manifold function h that gives the altitude at each point on the manifold. Let v be a vector field on the manifold, perhaps specifying a direction and rate of walking at every point on the manifold.

a. Form an expression that gives the power that must be expended to
follow the vector field at each point.

b. …

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Let f be a real-valued function, \mathbf{m} be a point, \mathbf{v} be a vector field on the manifold function respectively:

\begin{aligned} h \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) &= f\left(\chi (\mathbf{m}) \right) \\ \mathbf{v} &= \frac{d}{dt} \begin{bmatrix} x \\ y \end{bmatrix} \\ \end{aligned}

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\begin{aligned} P &= mg \frac{d h}{d t} \\ &= mg \left( \frac{\partial h}{\partial x} \frac{dx}{dt} + \frac{\partial h}{\partial y} \frac{dy}{dt} \right) \\ &= mg \begin{bmatrix} \frac{\partial h}{\partial x} & \frac{\partial h}{\partial y} \end{bmatrix} \begin{bmatrix} \frac{dx}{dt} \\ \frac{dy}{dt} \end{bmatrix} \\ &= mg (\nabla h) \cdot \mathbf{v} \\ &= mg (D f(\chi(\mathbf{m})) b(\chi{(\mathbf{m}})) \\ \end{aligned}

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— Me@2024-11-22 04:05:26 PM

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