Tag Archives: differentials

The Gelfand-Fuchs cocycle

Posted on 10 July, 2016 by jesperg

Let be a vector field on the Euclidean line . Expressed in a coordinate the vector field is . The logarithm of the component value at each point of the vector field is the fundamental Euclidean invariant We now let … Continue reading →

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The Lie bracket is torsion

Posted on 20 June, 2015 by jesperg

We start with a Lie group . An element acts on an external vector by . Let be generated by a vector at the identity , . By differentiating, with , the velocity of is This can also be interpreted … Continue reading →

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The partial derivative as a quotient

Posted on 11 March, 2015 by jesperg

Let . Then we have which of course is not possible to interpret as quotients. But if we change the notation to then this is possible to interpret as quotients. At a point on the function graph, the tangent plane defines a … Continue reading →

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Second derivative as a quotient

Posted on 9 February, 2015 by jesperg

The second derivative of a function can be expressed as So there is an extra correction term needed to be able to interpret the terms as quotients. To evaluate the second derivative we need to specify the velocity and acceleration … Continue reading →

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The second differential

Posted on 30 January, 2015 by jesperg

Let be a two-dimensional smooth manifold with local coordinates . For a real-valued function on this manifold we have the first differential The second differential is computed by applying to , and using Leibniz’s rule we get This is a … Continue reading →

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