A note on points of intersection

An interesting fact I thought about today. Let us suppose we have to determine the points of intersection of the cartesian equations and . What we generally do is . Why? Because the points of intersection will, on substitution, make both sides equal. On substitution, the values obtained on both sides will be . ButContinue reading “A note on points of intersection”

Directional derivative: Better explained than in Serge Lang’s book

This is an attempt to explain directional derivatives better than how it is explained in Serge Lang’s seminal book “A Second Course in Calculus”. The directional derivative of is grad , where is a unit vector in the direction that we’re interested in. Let us suppose we need to find the derivative of function alongContinue reading “Directional derivative: Better explained than in Serge Lang’s book”

A note on the gradient of a function.

I want to insert a note on (gradient of ). 1. It is not perpendicular to everything in the surface. Most proofs only go as far as to prove it is perpendicular to continuous parameterized curves. Nothing more. Stop reading too deeply into it. 2. It is mostly useful for finding perpendiculars to all parameterizedContinue reading “A note on the gradient of a function.”

The chain rule in multi-variable calculus: Generalized

Now we’ll discuss the chain rule for -nested functions. For example, an -nested function would be . What would be? We know that . If is continuous, then such that , which is equivalent to saying . In turn such that . Hence, we have Continuing like this, we get the formula such that forContinue reading “The chain rule in multi-variable calculus: Generalized”

Multi-variable differentiation.

There are very many bad books on multivariable calculus. “A Second Course in Calculus” by Serge Lang is the rare good book in this area. Succinct, thorough, and rigorous. This is an attempt to re-create some of the more orgasmic portions of the book. In space, should differentiation be defined as ? No, as divisionContinue reading “Multi-variable differentiation.”

Continuity decoded

The definition of continuity was framed after decades of deliberation and mathematical squabbling. The current notation we have is due to a Polish mathematician by the name of Weierstrass. It states that “If is continuous at point , then for every , such that for , .” Now let us try and interpret the statementContinue reading “Continuity decoded”

An attempted generalization of the Third Isomorphism Theorem.

I recently posted this question on math.stackexchange.com. The link is this. My assertion was “Let be a group with three normal subgroups and such that . Then . This is a generalization of the Third Isomorphism Theorem, which states that , where .” What was my rationale behind asking this question? Let be a groupContinue reading “An attempted generalization of the Third Isomorphism Theorem.”

Generalizing dual spaces- A study on functionals.

A functional is that which maps a vector space to a scalar field like or . If is the vector space under consideration, and (or ), then the vector space of functionals is referred to as the algebraic dual space . Similarly, the vector space of functionals (or ) is referred to as the secondContinue reading “Generalizing dual spaces- A study on functionals.”