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Question: Assuming that functions $f(x, y, z), U(x, y, z), V(x, y, z)$, and $W(x, y, z)$ have continuous second-order partial derivatives, i.e. the order of differentiation is not important. Given a scalar field $f(x, y, z)$. Prove that the gradient of field $f$ is an irrotational vector field.
| Step-by-Step Answer: Scalar field $f(x, y, z)$. Prove that gradient of field $f(i, e, \nabla f)$ is irrotational, or $\nabla \times(\nabla f)=0$
Proof:
$$
\nabla f=\left(f_{x}, f_{y}, f_{z}\right), \text { where } f_{x}=\frac{\partial f}{\partial x}, f_{y}=\frac{\partial f}{\partial y}, f_{z}=\frac{\partial f}{\partia... | $\nabla \times(\nabla f)=0$ or $\nabla f$ is irrotational | Scalar field $f(x, y, z)$. Prove that gradient of field $f(i, e, \nabla f)$ is irrotational, or $\nabla \times(\nabla f)=0$
Proof:
$$
\nabla f=\left(f_{x}, f_{y}, f_{z}\right), \text { where } f_{x}=\frac{\partial f}{\partial x}, f_{y}=\frac{\partial f}{\partial y}, f_{z}=\frac{\partial f}{\partial z}
$$
\[
\nabla \tim... |
Question: Determine $\operatorname{grad} \phi=\nabla \phi$ for the following scalar fields
$\phi(x, y, z)=e^{6 x y^3 z} \ln \left(4 x y^7 z^8\right)$
| Step-by-Step Answer: $$
\begin{aligned}
& \text { b) } \phi(x, y, z)=e^{6 x y^{3} z} \ln \left(4 x y^{7} z^{8}\right) \\
& \nabla \phi=\phi_{x} \cdot \hat{i}+\phi_{y} \hat{j}+\phi_{z} \hat{k},
\end{aligned}
$$
where $\phi_{x}=\frac{\partial \phi}{\partial x}=\left(e^{6 x y^{3} z}\right)_{x}^{\prime} \cdot \ln \left(4 x... | $\phi_{z} =6 x y^{3} \cdot e^{6 x y^{3} z} \cdot \ln \left(4 x y^{7} z^{8}\right)+\frac{8 e^{6 x y^{3} z}}{z}$ | $$
\begin{aligned}
& \text { b) } \phi(x, y, z)=e^{6 x y^{3} z} \ln \left(4 x y^{7} z^{8}\right) \\
& \nabla \phi=\phi_{x} \cdot \hat{i}+\phi_{y} \hat{j}+\phi_{z} \hat{k},
\end{aligned}
$$
where $\phi_{x}=\frac{\partial \phi}{\partial x}=\left(e^{6 x y^{3} z}\right)_{x}^{\prime} \cdot \ln \left(4 x y^{7} z^{8}\right)+e... |
Question: Calculate the Fourier transform of the two-sided exponential pulse given by
\[
f(t) =
\begin{cases}
e^{at}, & \text{if } t \leq 0 \\
e^{-at}, & \text{if } t > 0
\end{cases}, \quad a > 0
\] | Step-by-Step Answer: Fourice Transform:\\
\begin{tikzpicture}
\begin{axis}[
axis lines=middle,
xlabel={$t$},
ylabel={$f(t)$},
xtick=\empty,
ytick={1},
ymin=-0.1, ymax=1.5,
xmin=-3, xmax=3,
samples=200,
domain=-3:3,
width=12cm,
height=8cm,
every a... | $\frac{2 a}{a^{2}+\omega^{2}}$ | Fourice Transform:\\
\begin{tikzpicture}
\begin{axis}[
axis lines=middle,
xlabel={$t$},
ylabel={$f(t)$},
xtick=\empty,
ytick={1},
ymin=-0.1, ymax=1.5,
xmin=-3, xmax=3,
samples=200,
domain=-3:3,
width=12cm,
height=8cm,
every axis x label/.style={a... |
Question: Consider finding the root of $f(x)=x^2-3$. Let the tolerance tol $=0.01$ and start with the interval $[1,2]$, solve the equation using the Bisection method. | Step-by-Step Answer: Consider finding the root of $f(x)=x^{2}-3$. Let the tolerance fol $=0.01$ and start with the interval $[1,2]$, solve the equation using the Bisection method.
\section*{Solution:}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
$x_{1}$ & $x_{2}$ & $f\left(x_{1}\right)$ & $f\left(x_{2}\rig... | $1.7344$ | Consider finding the root of $f(x)=x^{2}-3$. Let the tolerance fol $=0.01$ and start with the interval $[1,2]$, solve the equation using the Bisection method.
\section*{Solution:}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
$x_{1}$ & $x_{2}$ & $f\left(x_{1}\right)$ & $f\left(x_{2}\right)$ & $x_{3}=\left(x... |
Question: Given two vectors $\vec{u}=\hat{\imath}+2 \hat{\jmath}+3 \hat{k}$ and $\vec{v}=4 \hat{\imath}+5 \hat{\jmath}+\widehat{6 k}$, compute: The vector projection of $\vec{v}$ onto $\vec{u}$ | Step-by-Step Answer: $$
\begin{aligned}
& \vec{u}=\hat{i}+2 \hat{j}+3 \hat{k}=(1,2,3) \\
& \vec{v}=4 \hat{i}+5 \hat{j}+6 \hat{k}=(4,5,6)
\end{aligned}
$$
\begin{tikzpicture}[scale=2,>=stealth]
% Define points
\coordinate (O) at (0,0);
\coordinate (U) at (2,0);
\coordinate (V) at (1,1.5);
\coordinate (P) at (1... | $\frac{16}{7}(1,2,3)$ | $$
\begin{aligned}
& \vec{u}=\hat{i}+2 \hat{j}+3 \hat{k}=(1,2,3) \\
& \vec{v}=4 \hat{i}+5 \hat{j}+6 \hat{k}=(4,5,6)
\end{aligned}
$$
\begin{tikzpicture}[scale=2,>=stealth]
% Define points
\coordinate (O) at (0,0);
\coordinate (U) at (2,0);
\coordinate (V) at (1,1.5);
\coordinate (P) at (1,0);
% Vectors
\d... |
Question: Given a vector field $\vec{F}(x, y, z)=(2 x y+z) \hat{\imath}+x^2 \hat{\jmath}+x \hat{k}$.
Evaluate $\int_C \vec{F} \cdot d \vec{r}$, where C is a line segment $\vec{r}(t)$ from point $P(1,-1,2)$ to $Q(2,2,3)$.
| Step-by-Step Answer:
$$
\left.\left.\begin{array}{rl}
\int_{c} \vec{F} \cdot d \vec{r} & =\phi\left(r\left(t_{2}\right)\right)-\phi\left(r\left(t_{1}\right)\right)=\phi(\text { at point } Q)-\phi(\text { at point } P) \\
& =\phi(2,2,3)-\phi(1,-1,2)
\end{array}\right)\left(2^{2} \cdot 2+2 \cdot 3\right)-\left(1^{2} \c... | $13$ | $$
\left.\left.\begin{array}{rl}
\int_{c} \vec{F} \cdot d \vec{r} & =\phi\left(r\left(t_{2}\right)\right)-\phi\left(r\left(t_{1}\right)\right)=\phi(\text { at point } Q)-\phi(\text { at point } P) \\
& =\phi(2,2,3)-\phi(1,-1,2)
\end{array}\right)\left(2^{2} \cdot 2+2 \cdot 3\right)-\left(1^{2} \cdot(-1)+1.2\right)\righ... |
Question: Using the Laplace transform table, determine the Laplace transform of the following functions:$f(t)=3 t^{2}+\sin (2 t)$
| Step-by-Step Answer: Laplace transform table
$$
\begin{aligned}
& \text { a) } f(t)=3 t^{2}+\sin (2 t) \\
& L\left\{3 t^{2}+\sin (2 t)\right\}=L\left\{3 t^{2}\right\}+L\{\sin (2 t)\} \\
& \left\{\begin{array}{l}
3 t^{2} \longrightarrow 3 \cdot \frac{2!}{s^{2+2}}=\frac{6}{s^{3}} \\
\sin (2 t) \rightarrow \frac{2}{s^{2}... | $\frac{6}{s^{3}}+\frac{2}{s^{2}+4}\right$ | Laplace transform table
$$
\begin{aligned}
& \text { a) } f(t)=3 t^{2}+\sin (2 t) \\
& L\left\{3 t^{2}+\sin (2 t)\right\}=L\left\{3 t^{2}\right\}+L\{\sin (2 t)\} \\
& \left\{\begin{array}{l}
3 t^{2} \longrightarrow 3 \cdot \frac{2!}{s^{2+2}}=\frac{6}{s^{3}} \\
\sin (2 t) \rightarrow \frac{2}{s^{2}+2^{2}}=\frac{2}{s^{2}... |
Question: Using the Laplace transform table, find the inverse Laplace transform of the following functions: $F(s)=\frac{s+2}{(s+2)^{2}+9}$
| Step-by-Step Answer: $\mathbb{L}^{-1}\left\{\frac{s+2}{(s+2)^{2}+9}\right\}$
For calculating $L^{-1}\left\{\frac{s+2}{(s+2)^{2}+9}\right\}$, we use $\cos (a t) \leftrightarrow \frac{s}{s^{2}+a^{2}}$ and $e^{\text {sot }} f(t) \leftrightarrow F\left(s-s_{0}\right)$, i.e., $L^{-1}\left\{\frac{s}{s^{2}+\underset{a}{2} \ti... | $L^{-1}\left\{\frac{s+2}{(s+2)^{2}+9}\right\}=e^{-2 t} \cdot \cos (3 t)$ | $\mathbb{L}^{-1}\left\{\frac{s+2}{(s+2)^{2}+9}\right\}$
For calculating $L^{-1}\left\{\frac{s+2}{(s+2)^{2}+9}\right\}$, we use $\cos (a t) \leftrightarrow \frac{s}{s^{2}+a^{2}}$ and $e^{\text {sot }} f(t) \leftrightarrow F\left(s-s_{0}\right)$, i.e., $L^{-1}\left\{\frac{s}{s^{2}+\underset{a}{2} \tilde{g}_{i}}\right\} ... |
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