A difficulty in understanding the n-dimensional second order derivative. The Next CEO of Stack OverflowA 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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A difficulty in understanding the n-dimensional second order derivative.
The Next CEO of Stack OverflowA 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.”
$begingroup$
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_1u_2 + u_2^2$ from where the term $u_1u_2$ 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
277214
$endgroup$
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_1u_2 + u_2^2$ from where the term $u_1u_2$ 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
277214
$endgroup$
1
$begingroup$
What is this from??
$endgroup$
– Randall
Mar 18 at 12:49
$begingroup$
$ u_1^2 + u_1u_2 + u_2^2$ @Randall from where the term $u_1u_2$ 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_1u_2 + u_2^2$ from where the term $u_1u_2$ 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
277214
$endgroup$
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_1u_2 + u_2^2$ from where the term $u_1u_2$ 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
277214
asked Mar 18 at 12:48
hopefullyhopefully
277214
edited Mar 18 at 13:02
hopefully
asked Mar 18 at 12:48
hopefullyhopefully
277214
asked Mar 18 at 12:48
hopefullyhopefully
277214
asked Mar 18 at 12:48
hopefullyhopefully
277214
277214
1
$begingroup$
What is this from??
$endgroup$
– Randall
Mar 18 at 12:49
$begingroup$
$ u_1^2 + u_1u_2 + u_2^2$ @Randall from where the term $u_1u_2$ 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_1u_2 + u_2^2$ @Randall from where the term $u_1u_2$ 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_1u_2 + u_2^2$ @Randall from where the term $u_1u_2$ comes?
$endgroup$
– hopefully
Mar 18 at 12:51
$begingroup$
$ u_1^2 + u_1u_2 + u_2^2$ @Randall from where the term $u_1u_2$ 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
$begingroup$
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 $bf p=(2,3)inrm dom(f)$, in other words: $bf u$ is a vector in the tangent space $T_bf p$. One has
$$q(bf u)=sum_i, >k=1^2 f_.ik(bf p)>u_iu_k=[u_1 u_2]left[matrix-6&6cr 6&30crright]left[matrixu_1cr u_2crright]=-6u_1^2+12u_1u_2+30 u_2^2 .$$
answered Mar 18 at 13:10
Christian BlatterChristian Blatter
175k8115327
$endgroup$
$begingroup$
what is $T_p$?
$endgroup$
– hopefully
Mar 18 at 13:12
add a comment |
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
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 $bf p=(2,3)inrm dom(f)$, in other words: $bf u$ is a vector in the tangent space $T_bf p$. One has
$$q(bf u)=sum_i, >k=1^2 f_.ik(bf p)>u_iu_k=[u_1 u_2]left[matrix-6&6cr 6&30crright]left[matrixu_1cr u_2crright]=-6u_1^2+12u_1u_2+30 u_2^2 .$$
answered Mar 18 at 13:10
Christian BlatterChristian Blatter
175k8115327
$endgroup$
$begingroup$
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 $bf p=(2,3)inrm dom(f)$, in other words: $bf u$ is a vector in the tangent space $T_bf p$. One has
$$q(bf u)=sum_i, >k=1^2 f_.ik(bf p)>u_iu_k=[u_1 u_2]left[matrix-6&6cr 6&30crright]left[matrixu_1cr u_2crright]=-6u_1^2+12u_1u_2+30 u_2^2 .$$
answered Mar 18 at 13:10
Christian BlatterChristian Blatter
175k8115327
$endgroup$
$begingroup$
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 $bf p=(2,3)inrm dom(f)$, in other words: $bf u$ is a vector in the tangent space $T_bf p$. One has
$$q(bf u)=sum_i, >k=1^2 f_.ik(bf p)>u_iu_k=[u_1 u_2]left[matrix-6&6cr 6&30crright]left[matrixu_1cr u_2crright]=-6u_1^2+12u_1u_2+30 u_2^2 .$$
answered Mar 18 at 13:10
Christian BlatterChristian Blatter
175k8115327
$endgroup$
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 $bf p=(2,3)inrm dom(f)$, in other words: $bf u$ is a vector in the tangent space $T_bf p$. One has
$$q(bf u)=sum_i, >k=1^2 f_.ik(bf p)>u_iu_k=[u_1 u_2]left[matrix-6&6cr 6&30crright]left[matrixu_1cr u_2crright]=-6u_1^2+12u_1u_2+30 u_2^2 .$$
answered Mar 18 at 13:10
Christian BlatterChristian Blatter
175k8115327
edited Mar 18 at 13:41
answered Mar 18 at 13:10
Christian BlatterChristian Blatter
175k8115327
answered Mar 18 at 13:10
Christian BlatterChristian Blatter
175k8115327
answered Mar 18 at 13:10
Christian BlatterChristian Blatter
175k8115327
175k8115327
$begingroup$
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_1u_2 + u_2^2$ @Randall from where the term $u_1u_2$ 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