## 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 , , where and are open in , such that are differentiable on their respective domains. Then is also differentiable on , with for all .

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

In the case of , we have that for all ,

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

Lemma 1 Let , be differentiable with and . Then,

Instead of , one can also use any closed interval of .

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

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 be any differentiable function of on , which is continuous on , and be Riemann integrable on intervals in its domain. Then,

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

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

Proof: We have

Rearranging the above completes the proof.

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 .

### Analytic geometry prerequisites

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

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

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

Proposition 2 The distance between a point and a straight line given by 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 , which we shall call . We use to denote the distance between and . One notices that is equal to , where is the angle between the straight line given in the proposition and the straight line connecting and . 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.

## Preliminary definitions

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

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

The velocity vector of at is the derivative

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

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

## Tangent line and osculating plane of a curve

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

Lemma 1 Let be continuous in . Then,

Proof: Trivial and left to the reader.

Theorem 1

Proof: We have that and by Proposition 2 that

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

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

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

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.

### 曲线的长度

，我们得到矢量函数 的可导，其导数为

：一个简单的计算。

References