Laplace Even Conservative B3Ed9F

1. **Problem Statement:** (a) Find the inverse Laplace transform of $$L(f) = \frac{2(e^{-s} - e^{-3s})}{s^2 - 4}$$. (b) (i) Obtain the even periodic extension of the given function $f(t)$ defined on $[0,2]$. (ii) Sketch the periodic extension for $0 \leq t \leq 5$. (c) Given vector field $$F(x,y,z) = zi + yj + xk$$, (i) Show $F$ is conservative. (ii) Find work done by $F$ along curve $$r'(t) = 3t i - 5t j + t^2 k, -1 \leq t \leq 1$$. --- 2. **Part (a) Inverse Laplace Transform:** - Recall the Laplace transform of $$\sinh(at)$$ is $$\frac{a}{s^2 - a^2}$$. - Here denominator is $$s^2 - 4 = s^2 - 2^2$$, so $a=2$. - The numerator has $$2(e^{-s} - e^{-3s})$$, which corresponds to time shifts. - Using the second shifting theorem: $$\mathcal{L}^{-1}\{e^{-as}F(s)\} = u(t-a)f(t-a)$$ where $u$ is the unit step. - Write: $$L(f) = 2\left(\frac{e^{-s}}{s^2 - 4} - \frac{e^{-3s}}{s^2 - 4}\right)$$ - Inverse transform: $$f(t) = 2\left(u(t-1)\sinh(2(t-1)) - u(t-3)\sinh(2(t-3))\right)$$ 3. **Part (b)(i) Even periodic extension:** - Original $f(t)$ is defined on $[0,2]$. - Even extension means $f(-t) = f(t)$. - Periodic extension with period $T=2$ repeats $f(t)$ for all $t$. - Define $f_{even}(t)$ on $[-2,2]$ by: $$f_{even}(t) = \begin{cases} f(t), & 0 \leq t \leq 2 \\ f(-t), & -2 \leq t < 0 \end{cases}$$ - Then extend periodically with period 2. 4. **Part (b)(ii) Sketch for $0 \leq t \leq 5$:** - Sketch $f(t)$ on $[0,2]$. - Repeat the even extension reflected on $[-2,0]$. - Repeat the pattern every 2 units to cover $[0,5]$. 5. **Part (c)(i) Show $F$ is conservative:** - A vector field $F = P i + Q j + R k$ is conservative if curl $F = 0$. - Compute curl: $$\nabla \times F = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right) i - \left(\frac{\partial R}{\partial x} - \frac{\partial P}{\partial z}\right) j + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) k$$ - Here $P = z$, $Q = y$, $R = x$. - Calculate: $$\frac{\partial R}{\partial y} = \frac{\partial x}{\partial y} = 0$$ $$\frac{\partial Q}{\partial z} = \frac{\partial y}{\partial z} = 0$$ $$\frac{\partial R}{\partial x} = \frac{\partial x}{\partial x} = 1$$ $$\frac{\partial P}{\partial z} = \frac{\partial z}{\partial z} = 1$$ $$\frac{\partial Q}{\partial x} = \frac{\partial y}{\partial x} = 0$$ $$\frac{\partial P}{\partial y} = \frac{\partial z}{\partial y} = 0$$ - So curl: $$\nabla \times F = (0 - 0) i - (1 - 1) j + (0 - 0) k = 0$$ - Hence $F$ is conservative. 6. **Part (c)(ii) Work done by $F$ along $C$:** - Work done $W = \int_C F \cdot dr$. - Since $F$ is conservative, $W = \phi(r(1)) - \phi(r(-1))$ where $\phi$ is potential function. - Find potential $\phi$ such that $\nabla \phi = F$: $$\frac{\partial \phi}{\partial x} = z$$ $$\frac{\partial \phi}{\partial y} = y$$ $$\frac{\partial \phi}{\partial z} = x$$ - Integrate $\frac{\partial \phi}{\partial y} = y$ w.r.t. $y$: $$\phi = \frac{y^2}{2} + g(x,z)$$ - Differentiate w.r.t. $x$: $$\frac{\partial \phi}{\partial x} = \frac{\partial g}{\partial x} = z$$ - So $g(x,z) = xz + h(z)$. - Differentiate $\phi$ w.r.t. $z$: $$\frac{\partial \phi}{\partial z} = x + h'(z) = x$$ - So $h'(z) = 0 \Rightarrow h(z) = C$ constant. - Potential function: $$\phi = \frac{y^2}{2} + xz + C$$ - Parametrize $r(t)$ by integrating $r'(t)$: $$r(t) = \int r'(t) dt = \int (3t, -5t, t^2) dt = \left(\frac{3t^2}{2}, -\frac{5t^2}{2}, \frac{t^3}{3}\right) + \mathbf{C}$$ - Assume $r(0) = 0$ for simplicity, so $C=0$. - Evaluate $\phi$ at $r(1)$: $$x = \frac{3(1)^2}{2} = \frac{3}{2}, y = -\frac{5(1)^2}{2} = -\frac{5}{2}, z = \frac{1^3}{3} = \frac{1}{3}$$ $$\phi(r(1)) = \frac{(-\frac{5}{2})^2}{2} + \frac{3}{2} \cdot \frac{1}{3} = \frac{\frac{25}{4}}{2} + \frac{3}{6} = \frac{25}{8} + \frac{1}{2} = \frac{25}{8} + \frac{4}{8} = \frac{29}{8}$$ - Evaluate $\phi$ at $r(-1)$: $$x = \frac{3(-1)^2}{2} = \frac{3}{2}, y = -\frac{5(-1)^2}{2} = -\frac{5}{2}, z = \frac{(-1)^3}{3} = -\frac{1}{3}$$ $$\phi(r(-1)) = \frac{(-\frac{5}{2})^2}{2} + \frac{3}{2} \cdot \left(-\frac{1}{3}\right) = \frac{25}{8} - \frac{1}{2} = \frac{25}{8} - \frac{4}{8} = \frac{21}{8}$$ - Work done: $$W = \phi(r(1)) - \phi(r(-1)) = \frac{29}{8} - \frac{21}{8} = \frac{8}{8} = 1$$ --- **Final answers:** (a) $$f(t) = 2\left(u(t-1)\sinh(2(t-1)) - u(t-3)\sinh(2(t-3))\right)$$ (b)(i) Even periodic extension defined by reflecting $f(t)$ about $t=0$ and repeating every 2 units. (b)(ii) Sketch repeats this pattern for $0 \leq t \leq 5$. (c)(i) $F$ is conservative since $\nabla \times F = 0$. (c)(ii) Work done $W = 1$.