Smooth functions | Singularity theory | Multivariable calculus
Critical point is a wide term used in many branches of mathematics. When dealing with functions of a real variable, a critical point is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero. When dealing with complex variables, a critical point is, similarly, a point in the function's domain where it is either not holomorphic or the derivative is equal to zero. Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient is undefined or is equal to zero. The value of the function at a critical point is a critical value. This sort of definition extends to differentiable maps between and a critical point being, in this case, a point where the rank of the Jacobian matrix is not maximal. It extends further to differentiable maps between differentiable manifolds, as the points where the rank of the Jacobian matrix decreases. In this case, critical points are also called bifurcation points. In particular, if C is a plane curve, defined by an implicit equation f (x,y) = 0, the critical points of the projection onto the x-axis, parallel to the y-axis are the points where the tangent to C are parallel to the y-axis, that is the points where In other words, the critical points are those where the implicit function theorem does not apply. The notion of a critical point allows the mathematical description of an astronomical phenomenon that was unexplained before the time of Copernicus. A stationary point in the orbit of a planet is a point of the trajectory of the planet on the celestial sphere, where the motion of the planet seems to stop before restarting in the other direction. This occurs because of a critical point of the projection of the orbit into the ecliptic circle. (Wikipedia).
From playlist Applications of Calculus
How to find and classify critical points of functions
Download the free PDF from http://tinyurl.com/EngMathYT This video shows how to calculate and classify the critical points of functions of two variables. The ideas involve first and second order derivatives and are seen in university mathematics.
From playlist Mathematics for Finance & Actuarial Studies 2
How to find + classify critical points of functions
Download the free PDF http://tinyurl.com/EngMathYT This video shows how to calculate and classify the critical points of functions of two variables. The ideas involve first and second order derivatives and are seen in university mathematics.
From playlist Several Variable Calculus / Vector Calculus
How to find critical points of functions
Download the free PDF from http://tinyurl.com/EngMathYT This is an example illustrating how to find and classify the critical points of functions of two variables. Such ideas rely on the second derivative test and are seen in university mathematics.
From playlist Mathematics for Finance & Actuarial Studies 2
Definition of critical numbers and two examples of how to find critical numbers for a polynomial and a rational function.
From playlist Calculus
Finding critical points of functions
Download the free PDF http://tinyurl.com/EngMathYT This is an example illustrating how to find and classify the critical points of functions of two variables. Such ideas rely on the second derivative test and are seen in university mathematics.
From playlist Several Variable Calculus / Vector Calculus
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From playlist Calculus
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From playlist Mathematics
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From playlist Calculus
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From playlist Daniel Rubin Show, Full episodes
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PROGRAM TIPPING POINTS IN COMPLEX SYSTEMS (HYBRID) ORGANIZERS: Partha Sharathi Dutta (IIT Ropar, India), Vishwesha Guttal (IISc, India), Mohit Kumar Jolly (IISc, India) and Sudipta Kumar Sinha (IIT Ropar, India) DATE: 19 September 2022 to 30 September 2022 VENUE: Ramanujan Lecture Hall an
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Beginners Guide to Critical Points in Calculus - Chris Tisdell Live Stream
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From playlist Calculus for Beginners