0
How to prove that differentiating in the space of smooth functions is a linear transformation?
Recall that a linear transformation T:U-->V is one such that
1) T(x+y)=T(x)+T(y) for any x,y in U
2) T(cx)=cT(x) for x in U and c in R
All you need to do is show that differentiation has these two properties, where the domain is C^(infinity). We shall consider smooth functions from R to R for simplicity, but the argument is analogous for functions from R^n to R^m. Let D by the differential operator.
D[(f+g)(x)] = [d/dx](f+g)(x) = lim(h-->0)[(f+g)(x+h)-(f+g)(x)]/h
= lim(h-->0)[f(x+h)+g(x+g)-f(x)-g(x)]/h
(since (f+g)(x) is taken to mean f(x)+g(x))
=lim(h-->0)[f(x+h)-f(x)]/h + lim(h-->0)[g(x+h) - g(x)]/h
since the sum of limits is the limit of the sums
=[d/dx]f(x) + [d/dx]g(x) = D[f(x)] + D[g(x)].
As for ths second criterion, D[(cf)(x)]=lim(h-->0)[(cf)(x+h)-(cf)(x)]/h
=lim(h-->0)[c[f(x+h)]-c[f(x)]]/h
since (cf)(x) is taken to mean c[f(x)]
=c[lim(h-->0)[f(x+h)-f(x)]/h] = c[d/dx]f(x) = cD[f(x)].
since constants can be factored out of limits.
Therefore the two criteria hold, and if you wished to prove this for the general case, you would simply apply the same procedure to the Jacobian matrices corresponding to Df.
What else can I help you with?
Do all of the functions of the cerebellum have to do with control of smooth muscle or glands?
Neither; the cerebellum functions in the excretory process which INCLUDES smooth muscle.
What is smooth linear motion?
Smooth linear motion refers to movement in a straight line without any sudden jerks or jolts. It is characterized by constant velocity and lack of abrupt changes in direction or speed. This type of motion is often achieved through systems like linear guides, rails, or belts.
What are 2 functions of the endoplasmic reticulum?
if it's the rough ER then it functions to transport protein but if its the smooth ER it functions as a site of lipid and steroid synthesis
What are the function of oxytocin?
The major functions of oxytocin have to do with smooth muscle contraction.
What functions does a smooth muscle perform in the human body?
Smooth muscles perform various functions in the human body, including regulating the movement of internal organs, controlling blood flow, and assisting in the digestion process.
What is the difference in board feet and linear feet?
Board feet is the unit measure of rough lumber. Linear feet is the unit measure of S4S lumber. (smooth 4 sides)
Smooth ER function?
Interesting the answer is to produce functions
How do you rigorously prove that the circumference of a circle can be measured in linear units?
The circumference of a circle is its boundary - it is a perimeter and therefore is a linear measure. Whether it is a smooth curve, as in the case of a circle, or a set of line segments meeting at vertices is irrelevant to its being linear.
Where is smooth muscle and what are its functions?
Smooth muscles are an involuntary non-striated muscles. It is divided into two sub-groups; the Single unit (unitary) and multiunit smooth muscle Hope this helps! :)
What part of the body is like the smooth er?
The liver is like the smooth ER in the body because it plays a role in detoxification and lipid metabolism, similar to the functions of the smooth ER in cells.
What functions is performed by the smooth endoplasmic reticulum inside its membrane?
It makes lipids and carbohydrates.
Where can one view images of a smooth endoplasmic reticulum?
To view images of smooth endoplasmic reticulum one could look for biology books. Smooth endoplasmic reticulum has functions with the production in several metabolic processes, it is "smooth" because it is not studded with ribosomes.
