Mathview's video: An Algebra of Differentials Part 1

What are differentials? Here we discuss differentials, not as vectors, but as elements of an additive algebra on certain open sets of real numbers. By choosing these sets carefully we show that addition of differentials is closed. How do you prove it? We use the following approach: A mapping between Real numbers and Differentials is constructed so as to preserve addition of the additive group of real numbers. Such a mapping is called a homomorphism. The mapping is found to be linear, continuous, bijective, and can be used to establish a closed additive group on differentials. Here differentials are taken to be real numbers in open epsilon (arbitrarily small) neighborhoods of zero.