Alex Notes
Chapter 4 Vector-Valued Functions
| Page | Description | Equation |
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| 262 |
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\[ \] |
| 146 261 |
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\[ \begin{align*} \vect{v}=\vect{c}'\left(t\right) &= \left[ \begin{matrix} dx_1/dt\\ \vdots\\ \dx_n/\dt\\ \end{matrix} \right]\\ &=\left(\frac{dx_1}{dt},\ldots,\frac{dx_n}{dt}\right)\\ &=\left(x_1',\ldots,x_n'\right) \end{align*} \] |
| 261 |
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\[ \vect{v}=\norm{\vect{v}}=\norm{\vect{c}'(t)} \] |
| 262 |
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\[ \begin{multline*} \vect{a}\left(t\right) =\frac{d\vect{v}}{dt} =\vect{c}''\left(t\right)\\ =x''\left(t\right)\vect{i} +y''\left(t\right)\vect{j} +z''\left(t\right)\vect{k} \end{multline*} \] |
| 263 |
$\vect{c}$ is a
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\[ \vect{c}^\prime\left(t_0\right)\neq\zeros \] |
| 263 |
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\[ \] |
| 264 |
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\[ \vect{F}=m\vect{a} \] |
| 265 |
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\[ \begin{align*} m\vect{r}'' &=-\frac{GmM}{r^3}\vect{r}\\ &=-\frac{GmM}{r^2}\hat{\vect{r}} \end{align*} \] |
| 265 |
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\[ \omega=\frac{v}{r} \] |
| 265 |
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\[ \vect{r}\left(t\right) =\left(r\cos{\omega}t,r\sin{\omega}t\right) \] |
| 265 |
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\[ \vect{a}\left(t\right) =r''\left(t\right)=-\omega^2\vect{r}\left(t\right) =-r\omega^2\hat{\vect{r}} \] |
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\[ \vect{F}_c=m\vect{a}=-mr\omega^2\hat{\vect{r}} \] | |
| 266 |
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\[ T=\frac{2\pi r}{v}=\frac{2\pi}{\omega} \] |
| 266 |
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\[ T^2=\frac{4\pi^2}{GM}r^3=K_sr^3 \] |
| 272 |
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\[ \vect{J}=\vect{r}\left(t\right) \times m\dot{\vect{r}}\left(t\right) =\vect{r}\times\vect{p} \] |
| 272 289 |
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\[ \frac{d\vect{c}}{dt}=0 \] |
| 275 |
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\[ \begin{align*} L(\vect{c}) &=\int_{t_0}^{t_1}\norm{\vect{c}'(t)}\dt\\ &=\int_{t_0}^{t_1} \sqrt{ \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2 }\dt\\ &=\int_{t_0}^{t_1} \sqrt{ \left[x'(t)\right]^2 +\left[y'(t)\right]^2 +\left[z'(t)\right]^2 }\dt \end{align*} \] |
| 279 |
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\[ \begin{align*} L(\vect{c}) &=\int_{t_0}^{t_1}\norm{\vect{c}'(t)}\dt\\ &=\int_{t_0}^{t_1} \sqrt{ \left[x_1'(t)\right]^2 +\cdots +\left[x_n'(t)\right]^2 }\dt \end{align*} \] |
| 276 |
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\[ \] |
| 278 |
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\[ \begin{align*} d\vect{s} &= \dx\vect{i} +\dy\vect{j} +\dz\vect{k}\\ &= \left( \frac{dx}{dt}\vect{i} +\frac{dy}{dt}\vect{j} +\frac{dz}{dt}\vect{k} \right)\dt \end{align*} \] |
| 278 |
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\[ \begin{align*} \ds &=\sqrt{\dx^2+\dy^2+\dz^2}\\ &=\sqrt{ \left(\frac{dx}{dt}\right)^2 +\left(\frac{dy}{dt}\right)^2 +\left(\frac{dz}{dt}\right)^2 }\dt \end{align*} \] |
| 279 |
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\[ s\left(t\right)=\int_a^t \norm{\vect{c}'(u)}\du \] |
| 279 |
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\[ \frac{ds}{dt}=s'(t)=\norm{\vect{c}'(t)} \] |
| 279 |
(Example 6)
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\[ \begin{align*} L(\vect{c}) &=\int_a^b\sqrt{1+\left(\frac{dy}{dx}\right)^2}\dx\\ &=\int_{x=a}^{x=b}\sqrt{dx^2+dy^2}\\ &=\int_{x=a}^{x=b}\ds\\ &=\int_{t_0}^{t_1}\ds\\ \end{align*} \] |
| 278 |
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\[ \text{arc length}=\int_{t_0}^{t_1}\ds\\ \] |
| 282 |
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\[ \vect{T} =\frac{d\vect{r}}{ds} =\frac{d\vect{r}/dt}{ds/dt} =\frac{\vect{v}}{\abs{\vect{v}}} \] |
| 282 |
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\[ \kappa =\abslr{\frac{d\vect{T}}{ds}} =\abslr{ \frac{d}{ds} \left(\frac{d\vect{r}}{ds}\right) } =\abslr{\frac{d^2\vect{r}}{ds^2}}\\ =\frac{1}{\abs{\vect{v}}}\abslr{\frac{d\vect{T}}{dt}} =\frac{1}{\abs{\vect{v}}} \abslr{ \frac{d}{dt} \left( \frac{\vect{v}}{\abs{\vect{v}}} \right) } \] |
| 283 |
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\[ \vect{N}=\frac{1}{\kappa}\frac{d\vect{T}}{ds} =\frac{d\vect{T}/dt}{\abs{d\vect{T}/dt}} \] |
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\[ \] | |
| 283 |
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\[ \vect{B}=\vect{T}\times\vect{N} \] |
| 282 |
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\[ \kappa =\abslr{\frac{d\vect{T}}{ds}} =\frac {\abs{\vect{v}\times\vect{a}}} {\abs{\vect{v}}^3} \] |
| 283 |
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\[ \tau=-\frac{d\vect{B}}{ds}\cdot\vect{N} =\frac{ \begin{vmatrix} \dot{x}&\dot{y}&\dot{z}\\ \ddot{x}&\ddot{y}&\ddot{z}\\ \dddot{x}&\dddot{y}&\dddot{z} \end{vmatrix} }{\abs{\vect{v}\times\vect{a}}^2} \] |
| 288 |
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\[ a=a_T\vect{T}+a_N\vect{N}\\ a_T=\frac{d}{dt}\abs{\vect{v}}\\ a_N=\kappa\abs{\vect{v}}^2 =\sqrt{\abs{\vect{a}}^2-\vect{a}_T^2} \] |
| 287 |
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\[ \begin{multline*} \vect{F}=\vect{F}\left(x,y,z\right) =-\nabla V =-\frac{mMG}{r^3}\vect{r}\\ = \left( \frac{-mMG}{r^3}x, \frac{-mMG}{r^3}y, \frac{-mMG}{r^3}z \right) \end{multline*} \] |
| 288 |
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\[ V=-\frac{mMG}{r} \] |
| 288 |
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\[ \vect{F}=-\nabla V=\frac{\varepsilon Qe}{r^3}\vect{r} \] |
| 288 |
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\[ \] |
| 289 |
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\[ E=\frac{1}{2}m\norm{\dot{\vect{r}}(t)}^2+V(\vect{r}(t)) \] |
| 289 |
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\[ \frac{dE}{dt}=0\\ \text{or}\\ E=\text{ constant} \] |
| 290 |
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\[ v_e =\sqrt{\frac{2MG}{R_0}} =\sqrt{2gR_0} \] |
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\[ G=6.67259 \left(85\right) \times10^{-11} \ \mathrm{N}\cdot\mathrm{m}^2/\mathrm{kg}^2 \] | |
| 290 |
$c(t)$ is a
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\[ \vect{c}^\prime\left(t\right) =\vect{F}\left(\vect{c}\left(t\right)\right) \] |
| 294 |
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\[ \begin{align*} \nabla &=\left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right)\\ &=\vect{i}\frac{\partial}{\partial x} +\vect{j}\frac{\partial}{\partial y} +\vect{k}\frac{\partial}{\partial z} \end{align*} \] |
| 294 |
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\[ \begin{align*} \nabla f &=\left(\frac{\partial f}{\partial x} ,\frac{\partial f}{\partial y} ,\frac{\partial f}{\partial z}\right)\\ &=\vect{i}\frac{\partial f}{\partial x} +\vect{j}\frac{\partial f}{\partial y} +\vect{k}\frac{\partial f}{\partial z} \end{align*} \] |
| 295 |
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\[ \text{div}\vect{F}=\nabla\cdot\vect{F}=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y} \] |
| 295 |
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\[ \text{div}\vect{F} =\nabla\cdot\vect{F} =\frac{\partial F_1}{\partial x} +\frac{\partial F_2}{\partial y} +\frac{\partial F_3}{\partial z} \] |
| 296 | See the book for more details about $V$ and this approximation. | \[ \left(\nabla\cdot\vect{F}\right) \left(\vect{x}_0\right)\approx \left. \frac{1}{V\left(0\right)} \frac{dV\left(t\right)}{dt} \right|_{t=0} \] |
| 295 | If $F$ is the velocity field of a fluid then it is expanding in volume. | \[ \nabla\cdot\vect{F}\lt0 \] |
| 295 | If $F$ is the velocity field of a fluid then it is neither expanding nor compressing. | \[ \nabla\cdot\vect{F}=0 \] |
| 295 | If $F$ is the velocity field of a fluid then it is compressing in volume. | \[ \nabla\cdot\vect{F}\gt0 \] |
| 299 |
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\[ \begin{multline*} \mathrm{curl}\ \vect{F} = \nabla\times\vect{F} = \begin{vmatrix} \vect{i}&\vect{j}&\vect{k}\\ \frac{\partial}{\partial x} &\frac{\partial}{\partial y} &\frac{\partial}{\partial z}\\ F_1&F_2&F_3 \end{vmatrix}\\ = \left( \frac{\partial F_3}{\partial y} -\frac{\partial F_2}{\partial z} \right)\vect{i} + \left( \frac{\partial F_1}{\partial z} -\frac{\partial F_3}{\partial x} \right)\vect{j}\\ + \left( \frac{\partial F_2}{\partial x} -\frac{\partial F_1}{\partial y} \right)\vect{k} \end{multline*} \] |
| 301 |
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\[ \vect{v}=\vect{v}\left(x,y,z\right) =\vect{\omega}\times\vect{r} =-\omega y\vect{i}+\omega x\vect{j}\\ \vect{r}=x\vect{i}+y\vect{j}+z\vect{k}\\ \vect{\omega}=\omega\vect{k} \] |
| 301 |
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\[ \nabla\times\vect{v}=2\vect{\omega} \] |
| 301 |
if this condition holds for the flow of a fluid
with velocity field $v$ then the vector field is
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\[ \nabla\times\vect{v}=\vect{0} \] |
| 303 | gradient is curl free for any $C^2$ function. | \[ \nabla\times\left(\nabla f\right)=\zeros \] |
| 304 | curl is divergence free for any $C^2$ function. | \[ \nabla\cdot\left(\nabla\times\vect{F}\right)=0 \] |
| 303 |
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\[ \nabla\times\vect{F}=\left(\frac{\partial F_1}{\partial x}-\frac{\partial F_2}{\partial x}\right)\vect{k} \] |
| 305 |
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\[ \begin{align*} \nabla^2 &= \left( \frac{\partial^2}{\partial x^2}, \frac{\partial^2}{\partial y^2}, \frac{\partial^2}{\partial z^2} \right)\\ &= \vect{i}\frac{\partial^2}{\partial x^2} +\vect{j}\frac{\partial^2}{\partial y^2} +\vect{k}\frac{\partial^2}{\partial z^2} \end{align*} \] |
| 305 |
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\[ \begin{align*} \nabla^2f &=\nabla\cdot\left(\nabla f\right)\\ &=\frac{\partial^2f}{\partial x^2} +\frac{\partial^2f}{\partial y^2} +\frac{\partial^2f}{\partial z^2} \end{align*} \] |
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