The Definition of Differentiability (Introduction and Example), Real Analysis II

In this video, I introduce the definition of differentiability for Real Analysis 2. We explore how differentiability for a function from R^n to R^m requires finding a linear transformation, denoted Df(p), that closely approximates the function near a point p. To this linear transformation we associate the action of an mxn matrix (also denoted Df(p), based on context). Dfferentiability indicates that the error between the actual function and its first-order approximation vanishes faster than the distance between the points.

We then connect this general definition to the familiar derivative from calculus by showing how the difference quotient for a function from R to R is a special case of the multivariable definition. The tangent line approximation in single-variable calculus is an example of this first-order approximation.

I walk through an example of a function from R^2 to R^3 and provide the associated matrix to show how to apply the definition. We see that the error between the function and its approximation drives the limit of the quotient to zero, proving the function is differentiable.

Finally, I discuss two alternative ways to express differentiability: one using epsilon-delta neighborhoods and the other expressing the function as its first-order approximation plus a small error term. Both formulations highlight the idea that differentiability captures how well the function can be approximated near a given point.

#Real Analysis #AdvancedCalculus #Differentiability #MultivariableCalculus #LinearTransformation #FirstOrderApproximation #Calculus #RealAnalysis #EpsilonDelta #MathematicsEducation #JacobianMatrix #differentiation

[MikeMagTech]

You are doing a great job teaching this! I am currently teaching myself tensor analysis and there is considerable overlap, which is an additional bonus.