Proving that $D^k_textbf{h}f(textbf{x}) =sum_{|alpha|=k}frac{k!}{alpha!}D^alpha f(textbf{x})h^alpha$
$begingroup$
I'm trying to prove this identity involving the k-th order directional derivative along a given vector $textbf{h}$:
$$D^k_textbf{h},f(textbf{x}) =sum_{|alpha|=k}frac{k!}{alpha!}D^alpha f(textbf{x})h^alpha,$$
using multi-index notation. (Here,
$D^k_textbf{h},f(textbf{x})equivsum^n_{i_1,...,i_k=1}D_{i_1,...,i_k}, f(textbf{x}),h_{i_1}...h_{i_k}$)
To begin with, I'd like to prove that
$$D^k_textbf{h},f(textbf{x})=(h_1D_1,f(textbf{x})+,...+h_nD_nf(textbf{x}))^k,$$
then I know from my lecture notes I can use the multinomial theorem to obtain the desired result. I'll proceed by induction. I know this equality holds for $k=1$, so I'll suppose it holds for $k$ and see if it is true for $k+1$. Using the induction hypothesis, I get
$$D^{k+1}_textbf{h},f(textbf{x})=D_textbf{h}(D^k_textbf{h},f(textbf{x}))=sum_{j=1}^nD_j(h_1D_1,f(textbf{x})+,...+h_nD_n,f(textbf{x}))^kh_j=(h_1D_1,f(textbf{x})+,...+h_nD_n,f(textbf{x}))^{k+1}.$$
I'm not sure of the last step though. I've used that, in general, $sum^n_{j=1}x_j(h_1x_1+,...+h_nx^n)^kh_j=(h_1x_1+,...+h_nx_n)^{k+1}$, but given that in this case the partial derivative is an operator, I'm having doubts about the applicability of this. Is the proof correct?
analysis multivariable-calculus
asked Jan 3 at 20:15
guest_userguest_user
946
$endgroup$
add a comment |
$begingroup$
I'm trying to prove this identity involving the k-th order directional derivative along a given vector $textbf{h}$:
$$D^k_textbf{h},f(textbf{x}) =sum_{|alpha|=k}frac{k!}{alpha!}D^alpha f(textbf{x})h^alpha,$$
using multi-index notation. (Here,
$D^k_textbf{h},f(textbf{x})equivsum^n_{i_1,...,i_k=1}D_{i_1,...,i_k}, f(textbf{x}),h_{i_1}...h_{i_k}$)
To begin with, I'd like to prove that
$$D^k_textbf{h},f(textbf{x})=(h_1D_1,f(textbf{x})+,...+h_nD_nf(textbf{x}))^k,$$
then I know from my lecture notes I can use the multinomial theorem to obtain the desired result. I'll proceed by induction. I know this equality holds for $k=1$, so I'll suppose it holds for $k$ and see if it is true for $k+1$. Using the induction hypothesis, I get
$$D^{k+1}_textbf{h},f(textbf{x})=D_textbf{h}(D^k_textbf{h},f(textbf{x}))=sum_{j=1}^nD_j(h_1D_1,f(textbf{x})+,...+h_nD_n,f(textbf{x}))^kh_j=(h_1D_1,f(textbf{x})+,...+h_nD_n,f(textbf{x}))^{k+1}.$$
I'm not sure of the last step though. I've used that, in general, $sum^n_{j=1}x_j(h_1x_1+,...+h_nx^n)^kh_j=(h_1x_1+,...+h_nx_n)^{k+1}$, but given that in this case the partial derivative is an operator, I'm having doubts about the applicability of this. Is the proof correct?
analysis multivariable-calculus
asked Jan 3 at 20:15
guest_userguest_user
946
$endgroup$
add a comment |
$begingroup$
I'm trying to prove this identity involving the k-th order directional derivative along a given vector $textbf{h}$:
$$D^k_textbf{h},f(textbf{x}) =sum_{|alpha|=k}frac{k!}{alpha!}D^alpha f(textbf{x})h^alpha,$$
using multi-index notation. (Here,
$D^k_textbf{h},f(textbf{x})equivsum^n_{i_1,...,i_k=1}D_{i_1,...,i_k}, f(textbf{x}),h_{i_1}...h_{i_k}$)
To begin with, I'd like to prove that
$$D^k_textbf{h},f(textbf{x})=(h_1D_1,f(textbf{x})+,...+h_nD_nf(textbf{x}))^k,$$
then I know from my lecture notes I can use the multinomial theorem to obtain the desired result. I'll proceed by induction. I know this equality holds for $k=1$, so I'll suppose it holds for $k$ and see if it is true for $k+1$. Using the induction hypothesis, I get
$$D^{k+1}_textbf{h},f(textbf{x})=D_textbf{h}(D^k_textbf{h},f(textbf{x}))=sum_{j=1}^nD_j(h_1D_1,f(textbf{x})+,...+h_nD_n,f(textbf{x}))^kh_j=(h_1D_1,f(textbf{x})+,...+h_nD_n,f(textbf{x}))^{k+1}.$$
I'm not sure of the last step though. I've used that, in general, $sum^n_{j=1}x_j(h_1x_1+,...+h_nx^n)^kh_j=(h_1x_1+,...+h_nx_n)^{k+1}$, but given that in this case the partial derivative is an operator, I'm having doubts about the applicability of this. Is the proof correct?
analysis multivariable-calculus
asked Jan 3 at 20:15
guest_userguest_user
946
$endgroup$
I'm trying to prove this identity involving the k-th order directional derivative along a given vector $textbf{h}$:
$$D^k_textbf{h},f(textbf{x}) =sum_{|alpha|=k}frac{k!}{alpha!}D^alpha f(textbf{x})h^alpha,$$
using multi-index notation. (Here,
$D^k_textbf{h},f(textbf{x})equivsum^n_{i_1,...,i_k=1}D_{i_1,...,i_k}, f(textbf{x}),h_{i_1}...h_{i_k}$)
To begin with, I'd like to prove that
$$D^k_textbf{h},f(textbf{x})=(h_1D_1,f(textbf{x})+,...+h_nD_nf(textbf{x}))^k,$$
then I know from my lecture notes I can use the multinomial theorem to obtain the desired result. I'll proceed by induction. I know this equality holds for $k=1$, so I'll suppose it holds for $k$ and see if it is true for $k+1$. Using the induction hypothesis, I get
$$D^{k+1}_textbf{h},f(textbf{x})=D_textbf{h}(D^k_textbf{h},f(textbf{x}))=sum_{j=1}^nD_j(h_1D_1,f(textbf{x})+,...+h_nD_n,f(textbf{x}))^kh_j=(h_1D_1,f(textbf{x})+,...+h_nD_n,f(textbf{x}))^{k+1}.$$
I'm not sure of the last step though. I've used that, in general, $sum^n_{j=1}x_j(h_1x_1+,...+h_nx^n)^kh_j=(h_1x_1+,...+h_nx_n)^{k+1}$, but given that in this case the partial derivative is an operator, I'm having doubts about the applicability of this. Is the proof correct?
analysis multivariable-calculus
analysis multivariable-calculus
asked Jan 3 at 20:15
guest_userguest_user
946
asked Jan 3 at 20:15
guest_userguest_user
946
asked Jan 3 at 20:15
guest_userguest_user
946
asked Jan 3 at 20:15
guest_userguest_user
946
asked Jan 3 at 20:15
guest_userguest_user
946
946
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3060953%2fproving-that-dk-textbfhf-textbfx-sum-alpha-k-frack-alphad%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3060953%2fproving-that-dk-textbfhf-textbfx-sum-alpha-k-frack-alphad%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
