Temperature Fields on the Euclidean Manifold $\mathbb{E}^{3}$

Eberhard Zeidler
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Eberhard Zeidler: Max Planck Institute for Mathematics in the Sciences

Chapter 10 in Quantum Field Theory III: Gauge Theory, 2011, pp 645-658 from Springer

Abstract: Abstract In mathematics and physics, differentiation describes the linearization of analytic objects like physical fields. In this chapter, let us study the directional derivative of a temperature field Θ on the Euclidean manifold $\mathbb{E}^{3}$ . To this end, let $$\varTheta: \mathbb{E}^3\to \mathbb{R}$$ be a smooth function. In terms of physics, we regard Θ(P) as the temperature at the point P on $\mathbb{E}^{3}$ . We are given the smooth curve $$C: P=P(t), \qquad t\in \mathbb{R}$$ on $\mathbb{E}^{3}$ with P 0:=P(0). In terms of position vectors at the origin, we describe the curve C by the smooth vector function x=x(t),t∈ℝ. The derivative $$\fbox{$d_{\mathbf{h}}\varTheta (P_{0}):= \frac{d\varTheta (\mathbf{x}(t))}{dt}_{|t=0}$}$$ is called the directional derivative of the temperature field Θ along the trajectory C at the point P 0.

Keywords: Velocity Vector; Real Line; Tangent Vector; Position Vector; Tangent Bundle (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/978-3-642-22421-8_11

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