What is integral of a scalar field with respect to a variable?
$begingroup$
I was reading Tom Apostol calculus volume 2 and came across theorem 10.8 (page 350)
The theorem is:
My doubt is, What is the definition of the right hand side of the underlined equation that is, What is the definition of $$int_{a}^b psi(vec x,t)dt $$, this integral formally? I know integral of real valued function formally to some extent? But what is the FORMAL DEFINITION OF THIS INTEGRAL?
real-analysis calculus functional-analysis multivariable-calculus vector-analysis
asked Dec 27 '18 at 17:46
Bijayan RayBijayan Ray
136112
$endgroup$
add a comment |
$begingroup$
I was reading Tom Apostol calculus volume 2 and came across theorem 10.8 (page 350)
The theorem is:
My doubt is, What is the definition of the right hand side of the underlined equation that is, What is the definition of $$int_{a}^b psi(vec x,t)dt $$, this integral formally? I know integral of real valued function formally to some extent? But what is the FORMAL DEFINITION OF THIS INTEGRAL?
real-analysis calculus functional-analysis multivariable-calculus vector-analysis
asked Dec 27 '18 at 17:46
Bijayan RayBijayan Ray
136112
$endgroup$
add a comment |
$begingroup$
I was reading Tom Apostol calculus volume 2 and came across theorem 10.8 (page 350)
The theorem is:
My doubt is, What is the definition of the right hand side of the underlined equation that is, What is the definition of $$int_{a}^b psi(vec x,t)dt $$, this integral formally? I know integral of real valued function formally to some extent? But what is the FORMAL DEFINITION OF THIS INTEGRAL?
real-analysis calculus functional-analysis multivariable-calculus vector-analysis
asked Dec 27 '18 at 17:46
Bijayan RayBijayan Ray
136112
$endgroup$
I was reading Tom Apostol calculus volume 2 and came across theorem 10.8 (page 350)
The theorem is:
My doubt is, What is the definition of the right hand side of the underlined equation that is, What is the definition of $$int_{a}^b psi(vec x,t)dt $$, this integral formally? I know integral of real valued function formally to some extent? But what is the FORMAL DEFINITION OF THIS INTEGRAL?
real-analysis calculus functional-analysis multivariable-calculus vector-analysis
real-analysis calculus functional-analysis multivariable-calculus vector-analysis
asked Dec 27 '18 at 17:46
Bijayan RayBijayan Ray
136112
asked Dec 27 '18 at 17:46
Bijayan RayBijayan Ray
136112
asked Dec 27 '18 at 17:46
Bijayan RayBijayan Ray
136112
asked Dec 27 '18 at 17:46
Bijayan RayBijayan Ray
136112
asked Dec 27 '18 at 17:46
Bijayan RayBijayan Ray
136112
136112
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You're confused because $psi(vec{x},,t)$ depends on multiple variables. But for each value of $vec{x}$, $psi(vec{x},,t)$ becomes just a function of $t$; you know how to integrate that. This recipe gives each $varphi(vec{x})$, thus defining the function $varphi$.
answered Dec 27 '18 at 17:59
J.G.J.G.
30.6k23149
$endgroup$
$begingroup$
I had thought of that about defining a different function g for each $vec x$, as a function of only t, then integrating g. But then how would you explain the next equation that is taking the partial derivative of $phi(vec x)$ ?
$endgroup$
– Bijayan Ray
Dec 28 '18 at 5:09
$begingroup$
@BijayanRag You have s function of $vec{x}$; differentiate it.
$endgroup$
– J.G.
Dec 28 '18 at 7:18
$begingroup$
your statement about function g (as per my last comment) is :- for all $vec x$ there exist a function g such that $psi (vec x,t)$=g(t)
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:38
$begingroup$
can you please tell me in formal words where are you getting a function of $vec x$ to differentiate?
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:42
$begingroup$
I think what I want to tell from the comments above is similar to the logical difference between uniform convergence and pointwise convergence & and also similar to logic behind existence of all directional derivative does not imply continuity
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:52
|
show 5 more comments
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%2f3054202%2fwhat-is-integral-of-a-scalar-field-with-respect-to-a-variable%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You're confused because $psi(vec{x},,t)$ depends on multiple variables. But for each value of $vec{x}$, $psi(vec{x},,t)$ becomes just a function of $t$; you know how to integrate that. This recipe gives each $varphi(vec{x})$, thus defining the function $varphi$.
answered Dec 27 '18 at 17:59
J.G.J.G.
30.6k23149
$endgroup$
$begingroup$
I had thought of that about defining a different function g for each $vec x$, as a function of only t, then integrating g. But then how would you explain the next equation that is taking the partial derivative of $phi(vec x)$ ?
$endgroup$
– Bijayan Ray
Dec 28 '18 at 5:09
$begingroup$
@BijayanRag You have s function of $vec{x}$; differentiate it.
$endgroup$
– J.G.
Dec 28 '18 at 7:18
$begingroup$
your statement about function g (as per my last comment) is :- for all $vec x$ there exist a function g such that $psi (vec x,t)$=g(t)
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:38
$begingroup$
can you please tell me in formal words where are you getting a function of $vec x$ to differentiate?
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:42
$begingroup$
I think what I want to tell from the comments above is similar to the logical difference between uniform convergence and pointwise convergence & and also similar to logic behind existence of all directional derivative does not imply continuity
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:52
|
show 5 more comments
$begingroup$
You're confused because $psi(vec{x},,t)$ depends on multiple variables. But for each value of $vec{x}$, $psi(vec{x},,t)$ becomes just a function of $t$; you know how to integrate that. This recipe gives each $varphi(vec{x})$, thus defining the function $varphi$.
answered Dec 27 '18 at 17:59
J.G.J.G.
30.6k23149
$endgroup$
$begingroup$
I had thought of that about defining a different function g for each $vec x$, as a function of only t, then integrating g. But then how would you explain the next equation that is taking the partial derivative of $phi(vec x)$ ?
$endgroup$
– Bijayan Ray
Dec 28 '18 at 5:09
$begingroup$
@BijayanRag You have s function of $vec{x}$; differentiate it.
$endgroup$
– J.G.
Dec 28 '18 at 7:18
$begingroup$
your statement about function g (as per my last comment) is :- for all $vec x$ there exist a function g such that $psi (vec x,t)$=g(t)
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:38
$begingroup$
can you please tell me in formal words where are you getting a function of $vec x$ to differentiate?
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:42
$begingroup$
I think what I want to tell from the comments above is similar to the logical difference between uniform convergence and pointwise convergence & and also similar to logic behind existence of all directional derivative does not imply continuity
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:52
|
show 5 more comments
$begingroup$
You're confused because $psi(vec{x},,t)$ depends on multiple variables. But for each value of $vec{x}$, $psi(vec{x},,t)$ becomes just a function of $t$; you know how to integrate that. This recipe gives each $varphi(vec{x})$, thus defining the function $varphi$.
answered Dec 27 '18 at 17:59
J.G.J.G.
30.6k23149
$endgroup$
You're confused because $psi(vec{x},,t)$ depends on multiple variables. But for each value of $vec{x}$, $psi(vec{x},,t)$ becomes just a function of $t$; you know how to integrate that. This recipe gives each $varphi(vec{x})$, thus defining the function $varphi$.
answered Dec 27 '18 at 17:59
J.G.J.G.
30.6k23149
answered Dec 27 '18 at 17:59
J.G.J.G.
30.6k23149
answered Dec 27 '18 at 17:59
J.G.J.G.
30.6k23149
answered Dec 27 '18 at 17:59
J.G.J.G.
30.6k23149
30.6k23149
$begingroup$
I had thought of that about defining a different function g for each $vec x$, as a function of only t, then integrating g. But then how would you explain the next equation that is taking the partial derivative of $phi(vec x)$ ?
$endgroup$
– Bijayan Ray
Dec 28 '18 at 5:09
$begingroup$
@BijayanRag You have s function of $vec{x}$; differentiate it.
$endgroup$
– J.G.
Dec 28 '18 at 7:18
$begingroup$
your statement about function g (as per my last comment) is :- for all $vec x$ there exist a function g such that $psi (vec x,t)$=g(t)
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:38
$begingroup$
can you please tell me in formal words where are you getting a function of $vec x$ to differentiate?
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:42
$begingroup$
I think what I want to tell from the comments above is similar to the logical difference between uniform convergence and pointwise convergence & and also similar to logic behind existence of all directional derivative does not imply continuity
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:52
|
show 5 more comments
$begingroup$
I had thought of that about defining a different function g for each $vec x$, as a function of only t, then integrating g. But then how would you explain the next equation that is taking the partial derivative of $phi(vec x)$ ?
$endgroup$
– Bijayan Ray
Dec 28 '18 at 5:09
$begingroup$
@BijayanRag You have s function of $vec{x}$; differentiate it.
$endgroup$
– J.G.
Dec 28 '18 at 7:18
$begingroup$
your statement about function g (as per my last comment) is :- for all $vec x$ there exist a function g such that $psi (vec x,t)$=g(t)
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:38
$begingroup$
can you please tell me in formal words where are you getting a function of $vec x$ to differentiate?
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:42
$begingroup$
I think what I want to tell from the comments above is similar to the logical difference between uniform convergence and pointwise convergence & and also similar to logic behind existence of all directional derivative does not imply continuity
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:52
$begingroup$
I had thought of that about defining a different function g for each $vec x$, as a function of only t, then integrating g. But then how would you explain the next equation that is taking the partial derivative of $phi(vec x)$ ?
$endgroup$
– Bijayan Ray
Dec 28 '18 at 5:09
$begingroup$
I had thought of that about defining a different function g for each $vec x$, as a function of only t, then integrating g. But then how would you explain the next equation that is taking the partial derivative of $phi(vec x)$ ?
$endgroup$
– Bijayan Ray
Dec 28 '18 at 5:09
$begingroup$
@BijayanRag You have s function of $vec{x}$; differentiate it.
$endgroup$
– J.G.
Dec 28 '18 at 7:18
$begingroup$
@BijayanRag You have s function of $vec{x}$; differentiate it.
$endgroup$
– J.G.
Dec 28 '18 at 7:18
$begingroup$
your statement about function g (as per my last comment) is :- for all $vec x$ there exist a function g such that $psi (vec x,t)$=g(t)
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:38
$begingroup$
your statement about function g (as per my last comment) is :- for all $vec x$ there exist a function g such that $psi (vec x,t)$=g(t)
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:38
$begingroup$
can you please tell me in formal words where are you getting a function of $vec x$ to differentiate?
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:42
$begingroup$
can you please tell me in formal words where are you getting a function of $vec x$ to differentiate?
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:42
$begingroup$
I think what I want to tell from the comments above is similar to the logical difference between uniform convergence and pointwise convergence & and also similar to logic behind existence of all directional derivative does not imply continuity
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:52
$begingroup$
I think what I want to tell from the comments above is similar to the logical difference between uniform convergence and pointwise convergence & and also similar to logic behind existence of all directional derivative does not imply continuity
$endgroup$
– Bijayan Ray
Dec 28 '18 at 10:52
|
show 5 more comments
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%2f3054202%2fwhat-is-integral-of-a-scalar-field-with-respect-to-a-variable%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
