Mixing
Connecting a point in space to a curve in xy plane
$begingroup$
Suppose that we have a point in space $p(1,2,3)$ and a curve in the $x-y$ plane, and suppose that $k(x,y)$ is a function that describes the unit vector in the direction of the tangent line to the curve $C:(x(t),y(t))$. Let S be the surface obtained by connecting $p$ to the points of $C$. Express the area of S as an integral of the first-type, of some function of x and y, using the function $k(x,y)$.
What I don't get is how can I use the function $k(x,y)$ in order to describe the surface area.
calculus integration geometry
edited Jan 7 at 17:17
David G. Stork
10.8k31432
asked Jan 7 at 17:13
alexander_yzalexander_yz
165
$endgroup$
add a comment |
$begingroup$
Suppose that we have a point in space $p(1,2,3)$ and a curve in the $x-y$ plane, and suppose that $k(x,y)$ is a function that describes the unit vector in the direction of the tangent line to the curve $C:(x(t),y(t))$. Let S be the surface obtained by connecting $p$ to the points of $C$. Express the area of S as an integral of the first-type, of some function of x and y, using the function $k(x,y)$.
What I don't get is how can I use the function $k(x,y)$ in order to describe the surface area.
calculus integration geometry
edited Jan 7 at 17:17
David G. Stork
10.8k31432
asked Jan 7 at 17:13
alexander_yzalexander_yz
165
$endgroup$
add a comment |
$begingroup$
Suppose that we have a point in space $p(1,2,3)$ and a curve in the $x-y$ plane, and suppose that $k(x,y)$ is a function that describes the unit vector in the direction of the tangent line to the curve $C:(x(t),y(t))$. Let S be the surface obtained by connecting $p$ to the points of $C$. Express the area of S as an integral of the first-type, of some function of x and y, using the function $k(x,y)$.
What I don't get is how can I use the function $k(x,y)$ in order to describe the surface area.
calculus integration geometry
edited Jan 7 at 17:17
David G. Stork
10.8k31432
asked Jan 7 at 17:13
alexander_yzalexander_yz
165
$endgroup$
Suppose that we have a point in space $p(1,2,3)$ and a curve in the $x-y$ plane, and suppose that $k(x,y)$ is a function that describes the unit vector in the direction of the tangent line to the curve $C:(x(t),y(t))$. Let S be the surface obtained by connecting $p$ to the points of $C$. Express the area of S as an integral of the first-type, of some function of x and y, using the function $k(x,y)$.
What I don't get is how can I use the function $k(x,y)$ in order to describe the surface area.
calculus integration geometry
calculus integration geometry
edited Jan 7 at 17:17
David G. Stork
10.8k31432
asked Jan 7 at 17:13
alexander_yzalexander_yz
165
edited Jan 7 at 17:17
David G. Stork
10.8k31432
asked Jan 7 at 17:13
alexander_yzalexander_yz
165
edited Jan 7 at 17:17
David G. Stork
10.8k31432
edited Jan 7 at 17:17
David G. Stork
10.8k31432
edited Jan 7 at 17:17
David G. Stork
10.8k31432
10.8k31432
asked Jan 7 at 17:13
alexander_yzalexander_yz
165
asked Jan 7 at 17:13
alexander_yzalexander_yz
165
asked Jan 7 at 17:13
alexander_yzalexander_yz
165
165
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Without loss of generality, suppose that we can write the curve as $y=f(x)$. Then $k(x,y)=f'(x)$ is the tangent to the curve. Let's choose a vector with components $Delta x$ and $Delta y$ You have $$k(x,y)=frac{Delta y}{Delta x}$$ Since we are talking about a vector in 3D space we will write this as $$v=(Delta x, k(x,y)Delta x,0)$$
You can write the area of the triangle formed by this vector and the vector from $p$ to $(x,y,0)$ as $$Delta A=frac 12|vtimes(p-(x,y,0))|$$
You can explicitly write out the cross product. Then sum up many of these areas (to get $S$), and put the limit $Delta xto 0$, and you will get an integral.
answered Jan 8 at 21:52
AndreiAndrei
11.8k21026
$endgroup$
$begingroup$
Thanks a lot! In regard to the first sentence, what if we cannot write the curve as y=f(x)?
$endgroup$
– alexander_yz
Jan 8 at 22:16
$begingroup$
The idea is the same, but you will need to write $Delta x$ in terms of $Delta t$, and integrate over that.
$endgroup$
– Andrei
Jan 8 at 22:21
$begingroup$
Great! You're really great :)
$endgroup$
– alexander_yz
Jan 8 at 22:35
add a comment |
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%2f3065245%2fconnecting-a-point-in-space-to-a-curve-in-xy-plane%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
$begingroup$
Without loss of generality, suppose that we can write the curve as $y=f(x)$. Then $k(x,y)=f'(x)$ is the tangent to the curve. Let's choose a vector with components $Delta x$ and $Delta y$ You have $$k(x,y)=frac{Delta y}{Delta x}$$ Since we are talking about a vector in 3D space we will write this as $$v=(Delta x, k(x,y)Delta x,0)$$
You can write the area of the triangle formed by this vector and the vector from $p$ to $(x,y,0)$ as $$Delta A=frac 12|vtimes(p-(x,y,0))|$$
You can explicitly write out the cross product. Then sum up many of these areas (to get $S$), and put the limit $Delta xto 0$, and you will get an integral.
answered Jan 8 at 21:52
AndreiAndrei
11.8k21026
$endgroup$
$begingroup$
Thanks a lot! In regard to the first sentence, what if we cannot write the curve as y=f(x)?
$endgroup$
– alexander_yz
Jan 8 at 22:16
$begingroup$
The idea is the same, but you will need to write $Delta x$ in terms of $Delta t$, and integrate over that.
$endgroup$
– Andrei
Jan 8 at 22:21
$begingroup$
Great! You're really great :)
$endgroup$
– alexander_yz
Jan 8 at 22:35
add a comment |
$begingroup$
Without loss of generality, suppose that we can write the curve as $y=f(x)$. Then $k(x,y)=f'(x)$ is the tangent to the curve. Let's choose a vector with components $Delta x$ and $Delta y$ You have $$k(x,y)=frac{Delta y}{Delta x}$$ Since we are talking about a vector in 3D space we will write this as $$v=(Delta x, k(x,y)Delta x,0)$$
You can write the area of the triangle formed by this vector and the vector from $p$ to $(x,y,0)$ as $$Delta A=frac 12|vtimes(p-(x,y,0))|$$
You can explicitly write out the cross product. Then sum up many of these areas (to get $S$), and put the limit $Delta xto 0$, and you will get an integral.
answered Jan 8 at 21:52
AndreiAndrei
11.8k21026
$endgroup$
$begingroup$
Thanks a lot! In regard to the first sentence, what if we cannot write the curve as y=f(x)?
$endgroup$
– alexander_yz
Jan 8 at 22:16
$begingroup$
The idea is the same, but you will need to write $Delta x$ in terms of $Delta t$, and integrate over that.
$endgroup$
– Andrei
Jan 8 at 22:21
$begingroup$
Great! You're really great :)
$endgroup$
– alexander_yz
Jan 8 at 22:35
add a comment |
$begingroup$
Without loss of generality, suppose that we can write the curve as $y=f(x)$. Then $k(x,y)=f'(x)$ is the tangent to the curve. Let's choose a vector with components $Delta x$ and $Delta y$ You have $$k(x,y)=frac{Delta y}{Delta x}$$ Since we are talking about a vector in 3D space we will write this as $$v=(Delta x, k(x,y)Delta x,0)$$
You can write the area of the triangle formed by this vector and the vector from $p$ to $(x,y,0)$ as $$Delta A=frac 12|vtimes(p-(x,y,0))|$$
You can explicitly write out the cross product. Then sum up many of these areas (to get $S$), and put the limit $Delta xto 0$, and you will get an integral.
answered Jan 8 at 21:52
AndreiAndrei
11.8k21026
$endgroup$
Without loss of generality, suppose that we can write the curve as $y=f(x)$. Then $k(x,y)=f'(x)$ is the tangent to the curve. Let's choose a vector with components $Delta x$ and $Delta y$ You have $$k(x,y)=frac{Delta y}{Delta x}$$ Since we are talking about a vector in 3D space we will write this as $$v=(Delta x, k(x,y)Delta x,0)$$
You can write the area of the triangle formed by this vector and the vector from $p$ to $(x,y,0)$ as $$Delta A=frac 12|vtimes(p-(x,y,0))|$$
You can explicitly write out the cross product. Then sum up many of these areas (to get $S$), and put the limit $Delta xto 0$, and you will get an integral.
answered Jan 8 at 21:52
AndreiAndrei
11.8k21026
answered Jan 8 at 21:52
AndreiAndrei
11.8k21026
answered Jan 8 at 21:52
AndreiAndrei
11.8k21026
answered Jan 8 at 21:52
AndreiAndrei
11.8k21026
11.8k21026
$begingroup$
Thanks a lot! In regard to the first sentence, what if we cannot write the curve as y=f(x)?
$endgroup$
– alexander_yz
Jan 8 at 22:16
$begingroup$
The idea is the same, but you will need to write $Delta x$ in terms of $Delta t$, and integrate over that.
$endgroup$
– Andrei
Jan 8 at 22:21
$begingroup$
Great! You're really great :)
$endgroup$
– alexander_yz
Jan 8 at 22:35
add a comment |
$begingroup$
Thanks a lot! In regard to the first sentence, what if we cannot write the curve as y=f(x)?
$endgroup$
– alexander_yz
Jan 8 at 22:16
$begingroup$
The idea is the same, but you will need to write $Delta x$ in terms of $Delta t$, and integrate over that.
$endgroup$
– Andrei
Jan 8 at 22:21
$begingroup$
Great! You're really great :)
$endgroup$
– alexander_yz
Jan 8 at 22:35
$begingroup$
Thanks a lot! In regard to the first sentence, what if we cannot write the curve as y=f(x)?
$endgroup$
– alexander_yz
Jan 8 at 22:16
$begingroup$
Thanks a lot! In regard to the first sentence, what if we cannot write the curve as y=f(x)?
$endgroup$
– alexander_yz
Jan 8 at 22:16
$begingroup$
The idea is the same, but you will need to write $Delta x$ in terms of $Delta t$, and integrate over that.
$endgroup$
– Andrei
Jan 8 at 22:21
$begingroup$
The idea is the same, but you will need to write $Delta x$ in terms of $Delta t$, and integrate over that.
$endgroup$
– Andrei
Jan 8 at 22:21
$begingroup$
Great! You're really great :)
$endgroup$
– alexander_yz
Jan 8 at 22:35
$begingroup$
Great! You're really great :)
$endgroup$
– alexander_yz
Jan 8 at 22:35
add a comment |
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%2f3065245%2fconnecting-a-point-in-space-to-a-curve-in-xy-plane%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
