A difficulty in understanding the n-dimensional second order derivative.A difficulty in understanding the...
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A difficulty in understanding the n-dimensional second order derivative.
A difficulty in understanding the proof of completeness of $l_{2}$.Difficulty (2) in understanding thm4.2 in Israel Gohberg.A difficulty in understanding Theorem 4.3 in Israel Gohberg.A difficulty in understanding the Gram determinant.A difficulty in understanding a part of a solution of 4.4.4 PetovicA difficulty in understanding a proof for L'Hospital's rule (in Petrovic)A difficulty in understanding a step in a solution.A difficulty in understanding a statement in example 10.6.6 Petrovic.A difficulty in understanding theorem 10.6.7 in Petrovic.(n-dimensional intermediate value theorem)A difficulty in understanding the definition of “Spaces of Matrix Elements.”
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The example and its solution is given below:
But I do not understand why in the calculation of $D^2 f(2,3)(u)^2$ the $u^2$ takes this form ....$ u_1^2 + u_{1}u_{2} + u_{2}^2$ from where the term $u_{1}u_{2}$ comes?...... could anyone explain this for me please?
Edit:
I remember that my professor said that the dot product in two dimensional is just the square of the term but I do not understand this statement..... but I do not understand this statement.
real-analysis calculus linear-algebra analysis multivariable-calculus
asked Mar 18 at 12:48
hopefullyhopefully
315215
$endgroup$
add a comment |
$begingroup$
The example and its solution is given below:
But I do not understand why in the calculation of $D^2 f(2,3)(u)^2$ the $u^2$ takes this form ....$ u_1^2 + u_{1}u_{2} + u_{2}^2$ from where the term $u_{1}u_{2}$ comes?...... could anyone explain this for me please?
Edit:
I remember that my professor said that the dot product in two dimensional is just the square of the term but I do not understand this statement..... but I do not understand this statement.
real-analysis calculus linear-algebra analysis multivariable-calculus
asked Mar 18 at 12:48
hopefullyhopefully
315215
$endgroup$
1
$begingroup$
What is this from??
$endgroup$
– Randall
Mar 18 at 12:49
$begingroup$
$ u_1^2 + u_{1}u_{2} + u_{2}^2$ @Randall from where the term $u_{1}u_{2}$ comes?
$endgroup$
– hopefully
Mar 18 at 12:51
$begingroup$
No, the book/notes.
$endgroup$
– Randall
Mar 18 at 12:52
$begingroup$
Petrovic "Advanced calculus theory and practice" @Randall
$endgroup$
– hopefully
Mar 18 at 12:54
$begingroup$
I remember that my professor said that the dot product in two dimensional is just the square of the term but I do not understand this statement.@Randall
$endgroup$
– hopefully
Mar 18 at 13:01
add a comment |
$begingroup$
The example and its solution is given below:
But I do not understand why in the calculation of $D^2 f(2,3)(u)^2$ the $u^2$ takes this form ....$ u_1^2 + u_{1}u_{2} + u_{2}^2$ from where the term $u_{1}u_{2}$ comes?...... could anyone explain this for me please?
Edit:
I remember that my professor said that the dot product in two dimensional is just the square of the term but I do not understand this statement..... but I do not understand this statement.
real-analysis calculus linear-algebra analysis multivariable-calculus
asked Mar 18 at 12:48
hopefullyhopefully
315215
$endgroup$
The example and its solution is given below:
But I do not understand why in the calculation of $D^2 f(2,3)(u)^2$ the $u^2$ takes this form ....$ u_1^2 + u_{1}u_{2} + u_{2}^2$ from where the term $u_{1}u_{2}$ comes?...... could anyone explain this for me please?
Edit:
I remember that my professor said that the dot product in two dimensional is just the square of the term but I do not understand this statement..... but I do not understand this statement.
real-analysis calculus linear-algebra analysis multivariable-calculus
real-analysis calculus linear-algebra analysis multivariable-calculus
asked Mar 18 at 12:48
hopefullyhopefully
315215
asked Mar 18 at 12:48
hopefullyhopefully
315215
edited Mar 18 at 13:02
hopefully
asked Mar 18 at 12:48
hopefullyhopefully
315215
asked Mar 18 at 12:48
hopefullyhopefully
315215
asked Mar 18 at 12:48
hopefullyhopefully
315215
315215
1
$begingroup$
What is this from??
$endgroup$
– Randall
Mar 18 at 12:49
$begingroup$
$ u_1^2 + u_{1}u_{2} + u_{2}^2$ @Randall from where the term $u_{1}u_{2}$ comes?
$endgroup$
– hopefully
Mar 18 at 12:51
$begingroup$
No, the book/notes.
$endgroup$
– Randall
Mar 18 at 12:52
$begingroup$
Petrovic "Advanced calculus theory and practice" @Randall
$endgroup$
– hopefully
Mar 18 at 12:54
$begingroup$
I remember that my professor said that the dot product in two dimensional is just the square of the term but I do not understand this statement.@Randall
$endgroup$
– hopefully
Mar 18 at 13:01
add a comment |
1
$begingroup$
What is this from??
$endgroup$
– Randall
Mar 18 at 12:49
$begingroup$
$ u_1^2 + u_{1}u_{2} + u_{2}^2$ @Randall from where the term $u_{1}u_{2}$ comes?
$endgroup$
– hopefully
Mar 18 at 12:51
$begingroup$
No, the book/notes.
$endgroup$
– Randall
Mar 18 at 12:52
$begingroup$
Petrovic "Advanced calculus theory and practice" @Randall
$endgroup$
– hopefully
Mar 18 at 12:54
$begingroup$
I remember that my professor said that the dot product in two dimensional is just the square of the term but I do not understand this statement.@Randall
$endgroup$
– hopefully
Mar 18 at 13:01
1
1
$begingroup$
What is this from??
$endgroup$
– Randall
Mar 18 at 12:49
$begingroup$
What is this from??
$endgroup$
– Randall
Mar 18 at 12:49
$begingroup$
$ u_1^2 + u_{1}u_{2} + u_{2}^2$ @Randall from where the term $u_{1}u_{2}$ comes?
$endgroup$
– hopefully
Mar 18 at 12:51
$begingroup$
$ u_1^2 + u_{1}u_{2} + u_{2}^2$ @Randall from where the term $u_{1}u_{2}$ comes?
$endgroup$
– hopefully
Mar 18 at 12:51
$begingroup$
No, the book/notes.
$endgroup$
– Randall
Mar 18 at 12:52
$begingroup$
No, the book/notes.
$endgroup$
– Randall
Mar 18 at 12:52
$begingroup$
Petrovic "Advanced calculus theory and practice" @Randall
$endgroup$
– hopefully
Mar 18 at 12:54
$begingroup$
Petrovic "Advanced calculus theory and practice" @Randall
$endgroup$
– hopefully
Mar 18 at 12:54
$begingroup$
I remember that my professor said that the dot product in two dimensional is just the square of the term but I do not understand this statement.@Randall
$endgroup$
– hopefully
Mar 18 at 13:01
$begingroup$
I remember that my professor said that the dot product in two dimensional is just the square of the term but I do not understand this statement.@Randall
$endgroup$
– hopefully
Mar 18 at 13:01
add a comment |
1 Answer
1
active
oldest
votes
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The object $q:=D^2f(2,3)$ is a quadratic form in the increment variable ${bf u}=(u_1,u_2)$. The vector ${bf u}$ is attached at the point ${bf p}=(2,3)in{rm dom}(f)$, in other words: ${bf u}$ is a vector in the tangent space $T_{bf p}$. One has
$$q({bf u})=sum_{i, >k=1}^2 f_{.ik}({bf p})>u_iu_k=[u_1 u_2]left[matrix{-6&6cr 6&30cr}right]left[matrix{u_1cr u_2cr}right]=-6u_1^2+12u_1u_2+30 u_2^2 .$$
answered Mar 18 at 13:10
Christian BlatterChristian Blatter
176k8115327
$endgroup$
$begingroup$
what is $T_{p}$?
$endgroup$
– hopefully
Mar 18 at 13:12
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
oldest
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$begingroup$
The object $q:=D^2f(2,3)$ is a quadratic form in the increment variable ${bf u}=(u_1,u_2)$. The vector ${bf u}$ is attached at the point ${bf p}=(2,3)in{rm dom}(f)$, in other words: ${bf u}$ is a vector in the tangent space $T_{bf p}$. One has
$$q({bf u})=sum_{i, >k=1}^2 f_{.ik}({bf p})>u_iu_k=[u_1 u_2]left[matrix{-6&6cr 6&30cr}right]left[matrix{u_1cr u_2cr}right]=-6u_1^2+12u_1u_2+30 u_2^2 .$$
answered Mar 18 at 13:10
Christian BlatterChristian Blatter
176k8115327
$endgroup$
$begingroup$
what is $T_{p}$?
$endgroup$
– hopefully
Mar 18 at 13:12
add a comment |
$begingroup$
The object $q:=D^2f(2,3)$ is a quadratic form in the increment variable ${bf u}=(u_1,u_2)$. The vector ${bf u}$ is attached at the point ${bf p}=(2,3)in{rm dom}(f)$, in other words: ${bf u}$ is a vector in the tangent space $T_{bf p}$. One has
$$q({bf u})=sum_{i, >k=1}^2 f_{.ik}({bf p})>u_iu_k=[u_1 u_2]left[matrix{-6&6cr 6&30cr}right]left[matrix{u_1cr u_2cr}right]=-6u_1^2+12u_1u_2+30 u_2^2 .$$
answered Mar 18 at 13:10
Christian BlatterChristian Blatter
176k8115327
$endgroup$
$begingroup$
what is $T_{p}$?
$endgroup$
– hopefully
Mar 18 at 13:12
add a comment |
$begingroup$
The object $q:=D^2f(2,3)$ is a quadratic form in the increment variable ${bf u}=(u_1,u_2)$. The vector ${bf u}$ is attached at the point ${bf p}=(2,3)in{rm dom}(f)$, in other words: ${bf u}$ is a vector in the tangent space $T_{bf p}$. One has
$$q({bf u})=sum_{i, >k=1}^2 f_{.ik}({bf p})>u_iu_k=[u_1 u_2]left[matrix{-6&6cr 6&30cr}right]left[matrix{u_1cr u_2cr}right]=-6u_1^2+12u_1u_2+30 u_2^2 .$$
answered Mar 18 at 13:10
Christian BlatterChristian Blatter
176k8115327
$endgroup$
The object $q:=D^2f(2,3)$ is a quadratic form in the increment variable ${bf u}=(u_1,u_2)$. The vector ${bf u}$ is attached at the point ${bf p}=(2,3)in{rm dom}(f)$, in other words: ${bf u}$ is a vector in the tangent space $T_{bf p}$. One has
$$q({bf u})=sum_{i, >k=1}^2 f_{.ik}({bf p})>u_iu_k=[u_1 u_2]left[matrix{-6&6cr 6&30cr}right]left[matrix{u_1cr u_2cr}right]=-6u_1^2+12u_1u_2+30 u_2^2 .$$
answered Mar 18 at 13:10
Christian BlatterChristian Blatter
176k8115327
edited Mar 18 at 13:41
answered Mar 18 at 13:10
Christian BlatterChristian Blatter
176k8115327
answered Mar 18 at 13:10
Christian BlatterChristian Blatter
176k8115327
answered Mar 18 at 13:10
Christian BlatterChristian Blatter
176k8115327
176k8115327
$begingroup$
what is $T_{p}$?
$endgroup$
– hopefully
Mar 18 at 13:12
add a comment |
$begingroup$
what is $T_{p}$?
$endgroup$
– hopefully
Mar 18 at 13:12
$begingroup$
what is $T_{p}$?
$endgroup$
– hopefully
Mar 18 at 13:12
$begingroup$
what is $T_{p}$?
$endgroup$
– hopefully
Mar 18 at 13:12
add a comment |
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1
$begingroup$
What is this from??
$endgroup$
– Randall
Mar 18 at 12:49
$begingroup$
$ u_1^2 + u_{1}u_{2} + u_{2}^2$ @Randall from where the term $u_{1}u_{2}$ comes?
$endgroup$
– hopefully
Mar 18 at 12:51
$begingroup$
No, the book/notes.
$endgroup$
– Randall
Mar 18 at 12:52
$begingroup$
Petrovic "Advanced calculus theory and practice" @Randall
$endgroup$
– hopefully
Mar 18 at 12:54
$begingroup$
I remember that my professor said that the dot product in two dimensional is just the square of the term but I do not understand this statement.@Randall
$endgroup$
– hopefully
Mar 18 at 13:01