June 6th, 2021

On the chain rule and change of variables of integrals

Originally published at 狗和留美者不得入内. You can comment here or there.

Theorem 1 (Chain rule) Let [公式] , [公式] , where [公式] and [公式] are open in [公式] , such that [公式] are differentiable on their respective domains. Then [公式] is also differentiable on [公式] , with [公式] for all [公式] .

Proof: We first assume that there exists a neighborhood [公式] of [公式] for which [公式] . This happens in the case of [公式] by inverse function theorem. In that case, by the definition of derivative and its properties, we have

[公式] In the case of [公式] , we have that for all [公式] ,

[公式]

From this, we easily verifies that [公式] , which means that [公式] is differentiable at [公式] and in the case of [公式] , [公式] must hold as well. [公式]

Lemma 1 Let [公式] , [公式] be differentiable [公式] with [公式] and [公式] . Then,

[公式]

Instead of [公式] , one can also use any closed interval of [公式] .

Proof: Follows directly from Fundamental Theorem of Calculus. See Theorem 2 (Newton-Leibniz axiom) of [1]. [公式]

Lemma 1 is a statement of invariance of integral along parameterized smooth paths with the same endpoints.

Theorem 2 (Change of variables or u-substitution in integration) Let [公式] be any differentiable function of [公式] on [公式] , which is continuous on [公式] , and [公式] be Riemann integrable on intervals in its domain. Then,

[公式]

Proof: Let [公式] be an antiderivative of [公式] . By the Fundamental Theorem of Calculus, it suffices to show that the left hand side of [公式] is equal to [公式] , which can be done by applying Lemma 1 accordingly. [公式]

Theorem 3 (Integration by parts) Let [公式] be differentiable functions on [公式] and continuous on [公式] . Then,

[公式]

Proof: We have

[公式]

Rearranging the above completes the proof. [公式]

References

On the tangent line and osculating plane of a curve

Originally published at 狗和留美者不得入内. You can comment here or there.

Here, we will be working in [公式] .

Analytic geometry prerequisites

Proposition 1 The distance between a point [公式] and the plane given by [公式] is [公式] .

Proof: A normal vector of the plane is [公式] . We plug in [公式] to get

[公式] the solution of which is [公式] . Since every unit of [公式] corresponds to [公式] of distance, we have for our answer[公式]. [公式]

Proposition 2 The distance between a point [公式] and a straight line given by [公式] can be obtained by the magnitude of a cross product.

Proof: As for this distance, it is obtained by taking the perpendicular with respect the straight line that contains [公式] , which we shall call [公式] . We use [公式] to denote the distance between [公式] and [公式] . One notices that [公式] is equal to [公式] , where [公式] is the angle between the straight line given in the proposition and the straight line connecting [公式] and [公式] . We know that the magnitude of the cross product of two vectors is the product of their magnitudes and the sign of the angle between the two vectors, which completes our proof. [公式]

Preliminary definitions

Definition 1 A regular curve is a connected subset [公式] of [公式] homeomorphic to some [公式] that is a line segment [公式] or a circle of radius [公式] . If the homeomorphism [公式] is in [公式] for [公式] and the rank of [公式] is maximal (equal to 1), then we say this curve is k-fold continuously differentiable. For [公式] , we say that [公式] is smooth.

Definition 2 Let a smooth curve [公式] be given by the parametric equations

[公式] The velocity vector of [公式] at [公式] is the derivative

[公式]The velocity vector field is the vector function [公式] . The speed of [公式] at [公式] is the length [公式] of the velocity vector.

Definition 3 The tangent line to a smooth curve [公式] at the point [公式] is the straight line through the point [公式] in the direction of the velocity vector [公式] .

Tangent line and osculating plane of a curve

We let [公式] denote the length of a chord of a curve joining the points [公式] and [公式] and [公式] denote the length of a perpendicular dropped from [公式] onto the tangent line to [公式] at the point [公式] .

Lemma 1 Let [公式] be continuous in [公式] . Then,

[公式]

Proof: Trivial and left to the reader. [公式]

Theorem 1

[公式]

Proof: We have that [公式] and by Proposition 2 that

[公式]

We have, using properties of limits and keeping Lemma 1 in mind in the process,

[公式]

[公式]

Definition 4 A plane [公式] is called an osculating plane to a curve [公式] at a point [公式] if

[公式]

Theorem 2 At each point [公式] of a regular curve [公式] of class [公式] where [公式] , there is an osculating plane [公式] , and the vectors [公式] are orthogonal to its unit normal vector [公式] .

Proof: Based on the following diagram from [1],

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<p><small>Originally published at <a href="https://gmachine1729.wpcomstaging.com/2021/06/06/on-the-tangent-line-and-osculating-plane-of-a-curve/">狗和留美者不得入内</a>. You can comment here or <a href="https://gmachine1729.wpcomstaging.com/2021/06/06/on-the-tangent-line-and-osculating-plane-of-a-curve/#comments">there</a>.</small></p><div class="RichText ztext Post-RichText"> <p>Here, we will be working in <img src="https://www.zhihu.com/equation?tex=%5Cmathbb%7BR%7D%5E3" alt="[公式]" data-formula="\mathbb{R}^3" /> .</p> <h3>Analytic geometry prerequisites</h3> <p><b>Proposition 1</b> The distance between a point <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29" alt="[公式]" data-formula="(x_1,y_1,z_1)" /> and the plane given by <img src="https://www.zhihu.com/equation?tex=ax%2Bby%2Bcz+%3D+d" alt="[公式]" data-formula="ax+by+cz = d" /> is <img src="https://www.zhihu.com/equation?tex=%5Cfrac%7Bd-%28ax_1%2Bby_1%2Bcz_1%29%7D%7B%5Csqrt%7Ba%5E2%2Bb%5E2%2Bc%5E2%7D%7D" alt="[公式]" data-formula="\frac{d-(ax_1+by_1+cz_1)}{\sqrt{a^2+b^2+c^2}}" /> .</p> <p><i>Proof</i>: A normal vector of the plane is <img src="https://www.zhihu.com/equation?tex=%28a%2Cb%2Cc%29" alt="[公式]" data-formula="(a,b,c)" /> . We plug in <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29+%2B+t%28a%2Cb%2Cc%29" alt="[公式]" data-formula="(x_1,y_1,z_1) + t(a,b,c)" /> to get</p> <p><img src="https://www.zhihu.com/equation?tex=a%28x_1%2Bat%29%2Bb%28y_1%2Bbt%29%2Bc%28z_1%2Bct%29+%3D+d%2C%5C%5C" alt="[公式]" data-formula="a(x_1+at)+b(y_1+bt)+c(z_1+ct) = d,\\" /> the solution of which is <img src="https://www.zhihu.com/equation?tex=t+%3D+%5Cfrac%7Bd-%28ax_1%2Bby_1%2Bcz_1%29%7D%7Ba%5E2%2Bb%5E2%2Bc%5E2%7D" alt="[公式]" data-formula="t = \frac{d-(ax_1+by_1+cz_1)}{a^2+b^2+c^2}" /> . Since every unit of <img src="https://www.zhihu.com/equation?tex=t" alt="[公式]" data-formula="t" /> corresponds to <img src="https://www.zhihu.com/equation?tex=%5Csqrt%7Ba%5E2%2Bb%5E2%2Bc%5E2%7D" alt="[公式]" data-formula="\sqrt{a^2+b^2+c^2}" /> of distance, we have for our answer<img src="https://www.zhihu.com/equation?tex=%5Cfrac%7Bd-%28ax_1%2Bby_1%2Bcz_1%29%7D%7B%5Csqrt%7Ba%5E2%2Bb%5E2%2Bc%5E2%7D%7D" alt="[公式]" data-formula="\frac{d-(ax_1+by_1+cz_1)}{\sqrt{a^2+b^2+c^2}}" />. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" data-formula="\square" /></p> <p><b>Proposition 2</b> The distance between a point <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29" alt="[公式]" data-formula="(x_1,y_1,z_1)" /> and a straight line given by <img src="https://www.zhihu.com/equation?tex=%28x_0%2Bat%2C+y_0%2Bbt%2C+z_0%2Bct%29" alt="[公式]" data-formula="(x_0+at, y_0+bt, z_0+ct)" /> can be obtained by the magnitude of a cross product.</p> <p><i>Proof</i>: As for this distance, it is obtained by taking the perpendicular with respect the straight line that contains <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29" alt="[公式]" data-formula="(x_1,y_1,z_1)" /> , which we shall call <img src="https://www.zhihu.com/equation?tex=h" alt="[公式]" data-formula="h" /> . We use <img src="https://www.zhihu.com/equation?tex=d" alt="[公式]" data-formula="d" /> to denote the distance between <img src="https://www.zhihu.com/equation?tex=%28x_0%2Cy_0%2Cz_0%29" alt="[公式]" data-formula="(x_0,y_0,z_0)" /> and <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29" alt="[公式]" data-formula="(x_1,y_1,z_1)" /> . One notices that <img src="https://www.zhihu.com/equation?tex=h" alt="[公式]" data-formula="h" /> is equal to <img src="https://www.zhihu.com/equation?tex=d+%5Csin+%5Ctheta" alt="[公式]" data-formula="d \sin \theta" /> , where <img src="https://www.zhihu.com/equation?tex=%5Ctheta" alt="[公式]" data-formula="\theta" /> is the angle between the straight line given in the proposition and the straight line connecting <img src="https://www.zhihu.com/equation?tex=%28x_0%2Cy_0%2Cz_0%29" alt="[公式]" data-formula="(x_0,y_0,z_0)" /> and <img src="https://www.zhihu.com/equation?tex=%28x_1%2Cy_1%2Cz_1%29" alt="[公式]" data-formula="(x_1,y_1,z_1)" /> . We know that the magnitude of the cross product of two vectors is the product of their magnitudes and the sign of the angle between the two vectors, which completes our proof. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" data-formula="\square" /></p> <h2>Preliminary definitions</h2> <p><b>Definition 1</b> A <i>regular curve</i> is a connected subset <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> of <img src="https://www.zhihu.com/equation?tex=%5Cmathbb%7BR%7D%5E3" alt="[公式]" data-formula="\mathbb{R}^3" /> homeomorphic to some <img src="https://www.zhihu.com/equation?tex=G" alt="[公式]" data-formula="G" /> that is a line segment <img src="https://www.zhihu.com/equation?tex=%5Ba%2Cb%5D" alt="[公式]" data-formula="[a,b]" /> or a circle of radius <img src="https://www.zhihu.com/equation?tex=1" alt="[公式]" data-formula="1" /> . If the homeomorphism <img src="https://www.zhihu.com/equation?tex=%5Cvarphi%3A+G+%5Cto+%5Cgamma" alt="[公式]" data-formula="\varphi: G \to \gamma" /> is in <img src="https://www.zhihu.com/equation?tex=C%5Ek" alt="[公式]" data-formula="C^k" /> for <img src="https://www.zhihu.com/equation?tex=k+%5Cgeq1" alt="[公式]" data-formula="k \geq1" /> and the rank of <img src="https://www.zhihu.com/equation?tex=%5Cvarphi" alt="[公式]" data-formula="\varphi" /> is maximal (equal to 1), then we say this curve is <i>k-fold continuously differentiable</i>. For <img src="https://www.zhihu.com/equation?tex=k%3D1" alt="[公式]" data-formula="k=1" /> , we say that <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> is <i>smooth</i>.</p> <p><b>Definition 2</b> Let a smooth curve <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> be given by the parametric equations</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cmathbf%7B%5Coverrightarrow%7Br%7D%7D+%3D+%5Cmathbf%7B%5Coverrightarrow%7Br%7D%7D%28t%29+%3D+x%28t%29%5Cmathbf%7B%5Coverrightarrow%7Bi%7D%7D%2By%28t%29%5Cmathbf%7B%5Coverrightarrow%7Bj%7D%7D%2Bz%28t%29%5Cmathbf%7B%5Coverrightarrow%7Bk%7D%7D.%5C%5C" alt="[公式]" data-formula="\mathbf{\overrightarrow{r}} = \mathbf{\overrightarrow{r}}(t) = x(t)\mathbf{\overrightarrow{i}}+y(t)\mathbf{\overrightarrow{j}}+z(t)\mathbf{\overrightarrow{k}}.\\" /> The <i>velocity vector</i> of <img src="https://www.zhihu.com/equation?tex=%5Cmathbf%7B%5Coverrightarrow%7Br%7D%7D%28t%29" alt="[公式]" data-formula="\mathbf{\overrightarrow{r}}(t)" /> at <img src="https://www.zhihu.com/equation?tex=t+%3D+t_0" alt="[公式]" data-formula="t = t_0" /> is the derivative</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cmathbf%7B%5Coverrightarrow%7Br%7D%7D%27%28t_0%29+%3D+x%27%28t_0%29%5Cmathbf%7B%5Coverrightarrow%7Bi%7D%7D%2By%27%28t_0%29%5Cmathbf%7B%5Coverrightarrow%7Bj%7D%7D%2Bz%27%28t_0%29%5Cmathbf%7B%5Coverrightarrow%7Bk%7D%7D.%5C%5C" alt="[公式]" data-formula="\mathbf{\overrightarrow{r}}'(t_0) = x'(t_0)\mathbf{\overrightarrow{i}}+y'(t_0)\mathbf{\overrightarrow{j}}+z'(t_0)\mathbf{\overrightarrow{k}}.\\" />The <i>velocity vector field</i> is the vector function <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t%29" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t)" /> . The <i>speed</i> of <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t%29" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}(t)" /> at <img src="https://www.zhihu.com/equation?tex=t+%3D+t_0" alt="[公式]" data-formula="t = t_0" /> is the length <img src="https://www.zhihu.com/equation?tex=%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C" alt="[公式]" data-formula="|\overrightarrow{\mathbf{r}}'(t_0)|" /> of the velocity vector.</p> <p><b>Definition 3</b> The <i>tangent line</i> to a smooth curve <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> at the point <img src="https://www.zhihu.com/equation?tex=P+%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29" alt="[公式]" data-formula="P = \overrightarrow{\mathbf{r}}(t_0)" /> is the straight line through the point <img src="https://www.zhihu.com/equation?tex=P+%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29+%5Cin+%5Cgamma" alt="[公式]" data-formula="P = \overrightarrow{\mathbf{r}}(t_0) \in \gamma" /> in the direction of the velocity vector <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t_0)" /> .</p> <h2>Tangent line and osculating plane of a curve</h2> <p>We let <img src="https://www.zhihu.com/equation?tex=d" alt="[公式]" data-formula="d" /> denote the length of a chord of a curve joining the points <img src="https://www.zhihu.com/equation?tex=P+%3D+%5Cgamma%28t_0%29" alt="[公式]" data-formula="P = \gamma(t_0)" /> and <img src="https://www.zhihu.com/equation?tex=P_1+%3D+%5Cgamma%28t_1%29" alt="[公式]" data-formula="P_1 = \gamma(t_1)" /> and <img src="https://www.zhihu.com/equation?tex=h" alt="[公式]" data-formula="h" /> denote the length of a perpendicular dropped from <img src="https://www.zhihu.com/equation?tex=P_1" alt="[公式]" data-formula="P_1" /> onto the tangent line to <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> at the point <img src="https://www.zhihu.com/equation?tex=P" alt="[公式]" data-formula="P" /> .</p> <p><b>Lemma 1</b> Let <img src="https://www.zhihu.com/equation?tex=%28x%28t%29%2Cy%28t%29%2Cz%28t%29%29" alt="[公式]" data-formula="(x(t),y(t),z(t))" /> be continuous in <img src="https://www.zhihu.com/equation?tex=t" alt="[公式]" data-formula="t" /> . Then,</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clim_%7Bt_1+%5Cto+t_0%7D+%5B%28x%28t_0%29%2Cy%28t_0%29%2Cz%28t_0%29%29+%5Ctimes+%28x%28t_1%29%2Cy%28t_1%29%2Cz%28t_1%29%29%5D+%3D+%5Coverrightarrow%7B%5Cmathbf%7B0%7D%7D.%5C%5C" alt="[公式]" data-formula="\lim_{t_1 \to t_0} [(x(t_0),y(t_0),z(t_0)) \times (x(t_1),y(t_1),z(t_1))] = \overrightarrow{\mathbf{0}}.\\" /></p> <p><i>Proof</i>: Trivial and left to the reader. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" data-formula="\square" /></p> <p><b>Theorem 1</b></p> <p><img src="https://www.zhihu.com/equation?tex=%5Clim_%7Bd+%5Cto+0%7D+%5Cfrac%7Bh%7D%7Bd%7D+%3D+%5Clim_%7Bt_1+%5Cto+t_0%7D+%5Cfrac%7Bh%7D%7Bd%7D+%3D+0.%5C%5C" alt="[公式]" data-formula="\lim_{d \to 0} \frac{h}{d} = \lim_{t_1 \to t_0} \frac{h}{d} = 0.\\" /></p> <p><i>Proof</i>: We have that <img src="https://www.zhihu.com/equation?tex=d+%3D+%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C" alt="[公式]" data-formula="d = |\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)|" /> and by Proposition 2 that</p> <p><img src="https://www.zhihu.com/equation?tex=h+%3D+%5Cleft%7C%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7D%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%7D%5Ctimes+%5B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%5D%5Cright%7C.%5C%5C" alt="[公式]" data-formula="h = \left|\frac{\overrightarrow{\mathbf{r}}'(t_0)}{|\overrightarrow{\mathbf{r}}'(t_0)|}\times [\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)]\right|.\\" /></p> <p>We have, using properties of limits and keeping Lemma 1 in mind in the process,</p> <p><img src="https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D+%5Clim_%7Bd+%5Cto+0%7D+%5Cfrac%7Bh%7D%7Bd%7D%26%3D%26%5Clim_%7Bt_1%5Cto+t_0%7D%5Cleft%7C%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7D%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%7D%5Ctimes+%5Cleft%5B%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7D%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%7D%5Cright%5D%5Cright%7C%5C%5C+%26%3D%26%5Clim_%7Bt_1+%5Cto+t_0%7D+%5Cfrac%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%5Ctimes+%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7D%7Bt_1+-+t_0%7D%7C%7D%7B+%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%7C%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_1%29-%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7D%7Bt_1+-+t_0%7D%7C%7D%5C%5C+%26%3D%26+%5Cfrac%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%5Ctimes+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%7D%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%7C%5E2%7D+%3D+0.+%5Cend%7Beqnarray%7D%5C%5C" alt="[公式]" data-formula="\begin{eqnarray} \lim_{d \to 0} \frac{h}{d}&amp;=&amp;\lim_{t_1\to t_0}\left|\frac{\overrightarrow{\mathbf{r}}'(t_0)}{|\overrightarrow{\mathbf{r}}'(t_0)|}\times \left[\frac{\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)}{|\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)|}\right]\right|\\ &amp;=&amp;\lim_{t_1 \to t_0} \frac{|\overrightarrow{\mathbf{r}}'(t_0)\times \frac{\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)}{t_1 - t_0}|}{ |\overrightarrow{\mathbf{r}}'(t_0)||\frac{\overrightarrow{\mathbf{r}}'(t_1)-\overrightarrow{\mathbf{r}}'(t_0)}{t_1 - t_0}|}\\ &amp;=&amp; \frac{|\overrightarrow{\mathbf{r}}'(t_0)\times \overrightarrow{\mathbf{r}}'(t_0)|}{|\overrightarrow{\mathbf{r}}'(t_0)|^2} = 0. \end{eqnarray}\\" /></p> <p><img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" data-formula="\square" /></p> <p><b>Definition 4</b> A plane <img src="https://www.zhihu.com/equation?tex=%5Calpha" alt="[公式]" data-formula="\alpha" /> is called an <i>osculating plane</i> to a curve <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> at a point <img src="https://www.zhihu.com/equation?tex=P+%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29" alt="[公式]" data-formula="P = \overrightarrow{\mathbf{r}}(t_0)" /> if</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clim_%7Bd%5Cto+0%7D+%5Cfrac%7Bh%7D%7Bd%5E2%7D+%3D+%5Clim_%7Bt_1+%5Cto+t_0%7D+%5Cfrac%7Bh%7D%7Bd%5E2%7D+%3D+0.%5C%5C" alt="[公式]" data-formula="\lim_{d\to 0} \frac{h}{d^2} = \lim_{t_1 \to t_0} \frac{h}{d^2} = 0.\\" /></p> <p><b>Theorem 2</b> At each point <img src="https://www.zhihu.com/equation?tex=P%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29" alt="[公式]" data-formula="P= \overrightarrow{\mathbf{r}}(t_0)" /> of a regular curve <img src="https://www.zhihu.com/equation?tex=%5Cgamma" alt="[公式]" data-formula="\gamma" /> of class <img src="https://www.zhihu.com/equation?tex=C%5Ek" alt="[公式]" data-formula="C^k" /> where <img src="https://www.zhihu.com/equation?tex=k+%5Cgeq+2" alt="[公式]" data-formula="k \geq 2" /> , there is an osculating plane <img src="https://www.zhihu.com/equation?tex=%5Calpha" alt="[公式]" data-formula="\alpha" /> , and the vectors <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%2C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%27%28t_0%29" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t_0),\overrightarrow{\mathbf{r}}''(t_0)" /> are orthogonal to its unit normal vector <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D" alt="[公式]" data-formula="\overrightarrow{\boldsymbol{\beta}}" /> .</p> <p><i>Proof</i>: Based on the following diagram from [1],</p> <figure data-size="normal"><noscript><img src="https://i2.wp.com/pic4.zhimg.com/v2-d4b39ec764b969ae82b0c141ec3e117f_b.jpg?w=327&#038;ssl=1" data-caption="" data-size="normal" data-rawwidth="327" data-rawheight="234" class="content_image" data-recalc-dims="1" /></noscript><img class="content_image lazy" src="data:;base64,<svg xmlns='http://www.w3.org/2000/svg' width='327' height='234'></svg>&#8221; width=&#8221;327&#8243; data-caption=&#8221;&#8221; data-size=&#8221;normal&#8221; data-rawwidth=&#8221;327&#8243; data-rawheight=&#8221;234&#8243; data-actualsrc=&#8221;<a href="https://pic4.zhimg.com/v2-d4b39ec764b969ae82b0c141ec3e117f_b.jpg&#038;#8221" rel="nofollow">https://pic4.zhimg.com/v2-d4b39ec764b969ae82b0c141ec3e117f_b.jpg&#038;#8221</a>; /></figure> <p>we have</p> <p><img src="https://www.zhihu.com/equation?tex=d+%3D+%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_1%29+-+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29%7C%2C+%5Cqquad+h+%3D+%7C%5Clangle+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_1%29+-+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29%2C+%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D%5Crangle%7C.+%5Cqquad+%281%29%5C%5C" alt="[公式]" data-formula="d = |\overrightarrow{\mathbf{r}}(t_1) - \overrightarrow{\mathbf{r}}(t_0)|, \qquad h = |\langle \overrightarrow{\mathbf{r}}(t_1) - \overrightarrow{\mathbf{r}}(t_0), \overrightarrow{\boldsymbol{\beta}}\rangle|. \qquad (1)\\" /> We first prove the existence of osculating plane, for which there are two cases:</p> <ol> <li><img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29+%5Ctimes+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%27%28t_0%29+%5Cneq+0" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t_0) \times \overrightarrow{\mathbf{r}}''(t_0) \neq 0" /> .</li> <li><img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29+%5Ctimes+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%27%28t_0%29+%3D+0" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t_0) \times \overrightarrow{\mathbf{r}}''(t_0) = 0" /> .</li> </ol> <p>In the first case, we simply define the unit vector <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D+%3D+%5Cfrac%7B%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29+%5Ctimes+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29%7D%7B%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29+%5Ctimes+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29%7C%7D" alt="[公式]" data-formula="\overrightarrow{\boldsymbol{\beta}} = \frac{\overrightarrow{\mathbf{r}}(t_0) \times \overrightarrow{\mathbf{r}}(t_0)}{|\overrightarrow{\mathbf{r}}(t_0) \times \overrightarrow{\mathbf{r}}(t_0)|}" /> and in the second case, take any <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D" alt="[公式]" data-formula="\overrightarrow{\boldsymbol{\beta}}" /> orthogonal to <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}'(t_0)" /> , which is non-zero by definition of regular curve. In both cases, we have</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clangle+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%2C+%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D%5Crangle+%3D+%5Clangle+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%27%28t_0%29%2C+%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D%5Crangle+%3D+0.+%5Cqquad+%282%29%5C%5C" alt="[公式]" data-formula="\langle \overrightarrow{\mathbf{r}}'(t_0), \overrightarrow{\boldsymbol{\beta}}\rangle = \langle \overrightarrow{\mathbf{r}}''(t_0), \overrightarrow{\boldsymbol{\beta}}\rangle = 0. \qquad (2)\\" /> Let <img src="https://www.zhihu.com/equation?tex=%5Calpha" alt="[公式]" data-formula="\alpha" /> be the plane passing through the point <img src="https://www.zhihu.com/equation?tex=P+%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29" alt="[公式]" data-formula="P = \overrightarrow{\mathbf{r}}(t_0)" /> and orthogonal to <img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D" alt="[公式]" data-formula="\overrightarrow{\boldsymbol{\beta}}" /> . By Taylor&#8217;s formula,</p> <p><img src="https://www.zhihu.com/equation?tex=%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_1%29+-+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%28t_0%29+%3D+%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%28t_1+-+t_0%29+%2B+%5Cfrac%7B1%7D%7B2%7D%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%27%28t_0%29%28t_1+-+t_0%29%5E2+%2B+%5Coverrightarrow%7B%5Cboldsymbol%7Bo%7D%7D%28%7Ct_1+-+t_0%7C%5E2%29.+%5Cqquad+%283%29%5C%5C" alt="[公式]" data-formula="\overrightarrow{\mathbf{r}}(t_1) - \overrightarrow{\mathbf{r}}(t_0) = \overrightarrow{\mathbf{r}}'(t_0)(t_1 - t_0) + \frac{1}{2}\overrightarrow{\mathbf{r}}''(t_0)(t_1 - t_0)^2 + \overrightarrow{\boldsymbol{o}}(|t_1 - t_0|^2). \qquad (3)\\" /> Applying <img src="https://www.zhihu.com/equation?tex=%282%29%2C%283%29" alt="[公式]" data-formula="(2),(3)" /> to <img src="https://www.zhihu.com/equation?tex=%281%29" alt="[公式]" data-formula="(1)" /> gives</p> <p><img src="https://www.zhihu.com/equation?tex=h+%3D+%7C%5Clangle+%5Coverrightarrow%7B%5Cboldsymbol%7Bo%7D%7D%28%7Ct_1+-+t_0%7C%5E2%29%2C+%5Coverrightarrow%7B%5Cboldsymbol%7B%5Cbeta%7D%7D%5Crangle%7C%2C+%5Cqquad+d+%3D%7C%5Coverrightarrow%7B%5Cmathbf%7Br%7D%7D%27%28t_0%29%28t_1-t_0%29%2B%5Coverrightarrow%7B%5Cboldsymbol%7Bo%7D%7D%28%7Ct_1-t_0%7C%29%7C.%5C%5C" alt="[公式]" data-formula="h = |\langle \overrightarrow{\boldsymbol{o}}(|t_1 - t_0|^2), \overrightarrow{\boldsymbol{\beta}}\rangle|, \qquad d =|\overrightarrow{\mathbf{r}}'(t_0)(t_1-t_0)+\overrightarrow{\boldsymbol{o}}(|t_1-t_0|)|.\\" /> From this, one can verify without difficulty that</p> <p><img src="https://www.zhihu.com/equation?tex=%5Clim_%7Bt_1+%5Cto+t_0%7D+%5Cfrac%7Bh%7D%7Bd%5E2%7D+%3D+0.%5C%5C" alt="[公式]" data-formula="\lim_{t_1 \to t_0} \frac{h}{d^2} = 0.\\" /> For the other part of this theorem, we simply uses <img src="https://www.zhihu.com/equation?tex=%281%29" alt="[公式]" data-formula="(1)" /> and <img src="https://www.zhihu.com/equation?tex=%283%29" alt="[公式]" data-formula="(3)" /> on the limit equal to zero to deduce the desired orthogonality relations. The details will be left to the reader as an exercise. <img src="https://www.zhihu.com/equation?tex=%5Csquare" alt="[公式]" data-formula="\square" /></p> <p><b>References</b></p> <ul> <li>[1] <a class=" wrap external" href="https://link.zhihu.com/?target=https%3A//gmachine1729.wpcomstaging.com/%25e5%2590%258d%25e5%258d%2595-c%25d0%25bf%25d0%25b8%25d1%2581%25d0%25ba%25d0%25b8-lists/%25e5%2590%258d%25e5%258d%2595-c%25d0%25bf%25d0%25b8%25d1%2581%25d0%25ba%25d0%25b8-lists-%25e6%2595%25b0%25e5%25ad%25a6%25e5%2592%258c%25e7%2589%25a9%25e7%2590%2586%25e4%25b9%25a6-math-and-physics-books/victor-andreevich-toponogov-vladimir-rovenski-differential-geometry-of-curves-and-surfaces_-a-concise-guide-birkha%25cc%2588user-boston-2005/" target="_blank" rel="noopener noreferrer">Victor Andreevich Toponogov, Vladimir Rovenski &#8211; Differential Geometry of Curves and Surfaces: A Concise Guide (2005)</a></li> </ul> </div>

关于曲线的长度和曲率

Originally published at 狗和留美者不得入内. You can comment here or there.

找到了一本俄罗斯人写的微分几何教材,当初看了觉得太枯燥,但近几天当弥补了不少自己的微积分和分析基础后却开始学进去了。我写的英语的数学文章的结果大多都是细节自己独立重证了。如果基本直接抄书,那写的意义可能也没那么大。我想如果我要抄书,那就把它翻译成中文吧,顺便也把中文的名词也给学会。对关于曲线的曲率的那段,我现在就准备这样做,当然如我当时翻译朗道的教科书一样,有的时候也会加一点我自己的想法或补点细节。

讲曲率之前,得先讲讲曲线的长度。

曲线的长度

[公式] 为某个曲线的闭弧, [公式] 为其参数化; [公式] 。我们注意到一个多边形线[公式][公式] )的一个由穿过某有序的有限的点集合 [公式] 的相邻的点的线段构成的曲线。一个多边形线 [公式] 是一个正则内接于曲线 [公式] 的多边形当存在线段 [公式] 的以点 [公式] 的满足 [公式] 的分割 [公式] 。对每个多边形线对应其长度 [公式] 。我们以 [公式] 标记所有正则内接于曲线 [公式] 的多边形线的集合

定义 1.4.1 一个连续曲线 [公式][公式] 被称为可求长曲线

定义 1.4.2 可求长曲线 [公式]长度定义为 [公式]

定理 1.4.1 光滑曲线的闭弧是可求长的,其长度为

[公式]

证明:相当繁琐的用到区间分割的不等式估计。为了时间的考虑暂时不过。此证明也大概率类似于黎曼积分的重要定理的证明。

任意曲线若其所有闭弧都是可求长被称为可求长曲线。对可求长曲线,可以定义基于每一个闭弧的长度的存在的所谓的弧长参数化。取任意点 [公式] 并联于 [公式] 参数 [公式] 的零值。为任意其他点 [公式] ,对应于等于的 [公式] 的弧长 [公式] 参数的值,若 [公式][公式] 之后我们给予其正符号 [公式] ,若 [公式][公式] 之前,我们给予其负符号 [公式] 。若 [公式] 有个光滑正则参数化 [公式] ,其弧长参数化也是光滑的,正则的。当考虑到符号,我们推导出弧长 [公式] 。函数 [公式] 是可导的, [公式] 。从而,存在反函数 [公式] ,其导数为

[公式]

曲线[公式] 的弧长度(或单元速度)参数化定义于公式

[公式][公式] ,我们得到矢量函数 [公式] 的可导,其导数为

[公式] 最后一个公式表明此弧长参数化的正则的。以弧长参数化 [公式] 来表示,切矢量 [公式] ,主法矢量 [公式] 和副法矢量 [公式] 的形式简单如下:

[公式]

第一个公式由 [公式] 成立,第二个成立于等式

[公式] 从这个,我们得到了 [公式] 之间的正交。最后一个公式成立于矢量 [公式] 的定义。

曲线的曲率

[公式][公式] 里的一个光滑的曲线。在此取一个点 [公式] ,另一个点 [公式] 。我们以 [公式] 标记在 [公式][公式] 的弧长,以 [公式] 标记 [公式][公式][公式] 切矢量 [公式][公式]

定义 1.6.1 极限

[公式]

若存在,叫做曲线 [公式] 在点 [公式]曲率

我们会将曲线 [公式] 在点 [公式] 的曲率标记以 [公式]

例子 1.6.1 (a) 若 [公式] 是直线,在 [公式] 的所有点,[公式][公式] 。(b) 若 [公式] 是半径为 [公式] 的圆,很容易得到圆的所有点的曲率都是 [公式]

定理 1.6.1[公式][公式] 正则曲线。在其所有点都有曲率。若 [公式] 是个 [公式] 的正则参数化,则 [公式] .
证明[公式][公式] 的弧长参数化,令 [公式] 。从而, [公式][公式] 为矢量 [公式] 之间的角度。由于 [公式][公式] ,故

[公式]

以这些,我们证明了定理的断言曲率存在的第一部分,并得以公式

[公式][公式] 为任意 [公式] 的正则参数化。利用 [公式] 做个计算会得到

[公式]

定理 1.6.2 在一个 [公式] 正则曲线 [公式] 的任意的点,密切平面的唯一存在的必要并且足够的条件是 [公式] 的曲率在此点不等于零。

证明:从定理1.6.1,我们可以看到曲率不等于零当且仅当 [公式] 之间不是平行的。在这种情况下,根据 [2] 里的 Theorem 2 的证明,只有一个密切平面存在。当曲率等于零时, [公式] ,故 [公式][公式] ,一个直线。显然,任何包括直线的(多个)平面都是其密切平面。 [公式]

定理 1.6.3 我们假设某个嵌入三维空间的平面包含所有曲线的点。令 [公式] 为任意在线段 [公式] 上的连续函数。曲率函数为 [公式][公式] 为弧长参数的曲线 [公式] 有唯一的存在(在由刚性运动而定义的等价关系下)。

证明:我们想寻求满足

[公式] 的函数 [公式] 。当我们不失去一般性地假设平面的法矢量为 [公式] ,从公式 [公式] ,我们能得到

[公式] 然后不难发觉当

[公式] 我们会得到

[公式][公式] 会给我们

[公式]

[公式] 为在初始点的切矢量 [公式][公式] 轴的角度。两个曲率一致的曲线的切矢量的变化也是一致的,则之间的切矢量的差是常矢量,位置的差也是长矢量。我们主要到在这里,切矢量的范数皆为 [公式] 。很明显当此俩曲率同等,可以将坐标轴做刚性变换(旋转,反射或平移)而将某一个曲线映射到另一个。证明的剩下细节留给阅读者。 [公式]

习题 1.7.3 (平面曲线的Frenet公式)证明公式

[公式] 同等于等式

[公式]

:一个简单的计算。

References