You could get these equations a lot more quickly using the relationship you outlined in the previous video. If you interpret multiplication by a complex number as a matrix transformation, then the two terms on the diagonal of the matrix must be equal, and the two terms on the antidiagonal must be the negative of one another. Then just apply these constraints to the general 2x2 Jacobian matrix and you get the Cauchy-Riemann equations.
I would like to ask why is it that the derivative from all directions have to be the same but in multivariable calculus you can have different derivative when approaching a point from different directions
this is an incomplete answer but suffice to say it cannot be simply answered. A multi variable function can have partial derivatives but that doesnt imply the function is itself differentiable. the notion of differentiablilty needs to meet more conditions for multi variable functions. remember partial derivatives only capture the behaviour of a function along 1 axis since by definition of partial derivatives you hold one variable as a fixed value while the other variable can change. geometrically for single variable variable functions you can view the derivative as a unique tangent line, for 2 variables you need to find a unique tangent plane for all points defined on f(x,y)
You could get these equations a lot more quickly using the relationship you outlined in the previous video. If you interpret multiplication by a complex number as a matrix transformation, then the two terms on the diagonal of the matrix must be equal, and the two terms on the antidiagonal must be the negative of one another. Then just apply these constraints to the general 2x2 Jacobian matrix and you get the Cauchy-Riemann equations.
great video! I saw many videos where the CR equations were used but I was struggling one where it explained the derivation. Cheers!
Thank you. Glad you found it helpful for understanding.
Awesome video, thanks!
I would like to ask why is it that the derivative from all directions have to be the same but in multivariable calculus you can have different derivative when approaching a point from different directions
this is an incomplete answer but suffice to say it cannot be simply answered. A multi variable function can have partial derivatives but that doesnt imply the function is itself differentiable. the notion of differentiablilty needs to meet more conditions for multi variable functions. remember partial derivatives only capture the behaviour of a function along 1 axis since by definition of partial derivatives you hold one variable as a fixed value while the other variable can change. geometrically for single variable variable functions you can view the derivative as a unique tangent line, for 2 variables you need to find a unique tangent plane for all points defined on f(x,y)
Me a computer engineering major just wanting to know how to solve complex derivatives and limits haha, (I appreciate the vids)
thanks.
Ben1994?
Name was changed from my first name to one of my middle names.
mojo