He should.

## real analysis: why is the total differential defined for functions in an open set \$ U in mathbb {R ^ {n}} \$

This is a follow-up question to my previous question.

Definition: a function $$f: D to mathbb {R}$$ where $$D subset > mathbb {R}$$ is differentiable at a limit point $$z in D$$ Yes

$$lim limits_ {h to 0} frac {f (z + h) -f (z)} {h}, h neq 0, z + h in D$$
there then we say
$$f & # 39; (z)$$ is the derivative of $$f$$ to $$x$$.

Note that the conditions $$h neq 0, z + h in D$$ are necessary to ensure that the difference ratio is defined and $$z$$ it needs to be a limit point of $$D$$ such is possible to find a sequence that approaches $$z$$ other than the constant sequence (for which the difference quotient is not defined). If this were not the case, the limit would not be unique.

It is also easy to show from this definition that this is equal to the existence of $$c = f & # 39; (z) in mathbb {R}$$ such that $$forall h: x + h in D$$:

$$f (z + h) = f (z) + ch + r_ {z} (h)$$ with $$lim limits_ {h to 0} frac {r_ {z} (h)} {h} = 0$$.

Note that $$f & # 39; (z) h$$ it is a linear map defined for all $$h: z + h in D$$. Conditions again $$h neq 0, z + h in D$$ are necessary for the expression to make sense and $$z$$ it must be a limit point for $$c$$ be unqiue

The definition of limit is also easy to generalize to functions with vector values, but it makes no sense for functions of more than one variable. So, the idea is to define the differentiability in terms of the existence of a linear map.

Definition: Let $$U subset mathbb {R ^ {n}}$$ be open. A function $$f: U to mathbb {R ^ {}}$$ is differentiable in
$$z in U$$ if there is a linear map $$L: mathbb {R ^ {n}} to > mathbb {R m}$$ such that $$forall h: x + h in U$$

$$f (z + h) = f (z) + L_ {z} (h) + r_ {z} (h)$$ with $$lim limits_ {h a 0} > frac {r_ {z} (h)} { | h |} = 0$$.

This definition is taken from some class notes on real analysis and the linear map. $$L$$ can be considered as the best approximation of the change in the value of the function $$f$$ close $$z$$.

It seems clear to me why we come up with the idea of ​​differentiability as linear approximation. However, what I really don't understand is why we require $$U$$ be open. Of course, we need to be able to approach a point so that it is differentiable, but I feel that the condition is stronger than necessary. In particular, we did not need it for the case of functions of a variable. Can we not reuse the same conditions as in the case of a variable?

Another thing that bothers us is that we require $$L$$ to be defined in $$mathbb {R ^ n}$$. This also seems to be unnecessary and it would be sufficient to require it to be defined for $$h: x + h in U$$.

Am I missing something? Any clue is appreciated. Thank you!

## [ Politics ] Open question: Are Ashley Judd's liberal speeches useful or harmful to the Democratic Party?

When I saw her at the women's march, I thought she was having a seizure.

## Catalina – How to open the profile management window every time you open Chrome?

Chrome allows multiple people to use Chrome on the same computer, and each profile has its own bookmarks, settings and custom accounts. By default, Chrome opens in the profile used the last time the browser was opened.

I don't want that to happen. I want to see this screen every time I start Chrome

It is easy to do on Windows and Linux

How do I do it on my Mac?

## [ Politics ] Open question: Please help me. I am 24 years old and I have wasted the last 14 years of my life. What I can do?

[Policy] Open question: Please help me. I am 24 years old and I have wasted the last 14 years of my life. What I can do?

## [ Gender Studies ] Open question: Why aren't there many women pedophiles?

There are only men here who rape, kill or abuse children. Why are there more men than women who do this?

## [ Politics ] Open question: Why did a conservative cross the street?

to reach the WASP nest

## macos: quick shortcut (er) to open the link in a window or private tab in Safari

Is there a keyboard + click shortcut available in macOS Safari so you can immediately open a link in a private window when you click? I just want to hold down a key (or key combination) and then click and go without using the context menu.

Note: I Do not do means hold down the ⌥ key (option / alt) and select `Open Link in New Private Window` from the context menu (as answered in this question).

If it is not available by default in macOS, how can I create this solution with Automator or terminal commands (for example, maybe even using `defaults write ...`)?

## Oracle – ORA-01000: maximum open cursors exceeded golang

I have a service that runs a stack. For this, it performs the reading of this stack through a query in Oracle. It turns out that, after hours of execution, I receive the message:

ORA-01000: maximum open cursors exceeded

I wonder what could be happening. I have a global variable that opens the connection and when I run a query I always verify its connection and copy it to the local variable.

In Rows.Next (), so I read, when checking if there are no more lines, close the cursor, so I could not have accumulated so many open cursors to exceed the limit.

Bank connection:

``````type GERENCIACON struct {
DataBase *sql.DB
}

func (gc *GERENCIACON) F_FECHAR_CONEXAO() {
gc.DataBase.Close()
}

func (gc *GERENCIACON) F_ABRIR_CONEXAO()  {
if gc.DataBase == nil || gc.DataBase.Ping() != nil {
gc.DataBase, _ = sql.Open("goracle", "X/X@10.0.254.10:1521/orcl")
}
}

var VGGerenciaConexao GERENCIACON
``````

Structure for consultation:

``````type GERENCIACONSULTA struct {
DataBase *sql.DB
Rows *sql.Rows
}

func (gc *GERENCIACONSULTA) F_EXECUTA_CONSULTA(pSql string) {
VGGerenciaConexao.F_ABRIR_CONEXAO()
gc.DataBase = VGGerenciaConexao.DataBase
gc.Rows, _ = gc.DataBase.Query(pSql)
}
``````

Pile Execution Service

`````` var vGerenciaConsulta CertanoLabsPackage.GERENCIACONSULTA
var vSQL string

for {
vSQL = "select * from stack "

vGerenciaConsulta.F_EXECUTA_CONSULTA(vSQL)

for vGerenciaConsulta.Rows.Next() {
...
}

time.Sleep(time.Minute)
}
``````

Could someone tell me what could be happening?