2.5 Series, Hyperbolic Functions, Multivariable Calculus, and Laplace Transforms

2.5 Series, Hyperbolic Functions, Multivariable Calculus, and Laplace Transforms

The 2022 AMSTHC table of specifications (TOS) gives Calculus and Differential Equations five competencies each, at one item per competency. Section 2.3 covers derivatives and single-variable integration, and Section 2.4 covers first-order and constant-coefficient equations. This section covers the remaining competencies:

• Plane curves and hyperbolic functions (Calculus 1.3)
• Infinite sequences, power series and infinite series (Calculus 1.4)
• Partial derivatives (Calculus 1.5)
• Integration for areas, volumes, force and work (Differential Equations 2.1)
• Systems of linear differential equations (Differential Equations 2.3)
• Double and triple integrals (Differential Equations 2.4)
• Laplace transforms (Differential Equations 2.5)

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Plane Curves and Curvature

A curve may be given as $y = f(x)$, parametrically as $x = x(t)$ and $y = y(t)$, or in polar form as $r = f(\theta)$. Engineers most often need its slope, its arc length and its curvature.

Example. For $y = x^2/40$ at $x = 10$: $y' = 0.5$ and $y'' = 0.05$. Then $\rho = (1.25)^{1.5}/0.05 = 27.95$. This is the same idea as the radius of a parabolic vertical curve at a point.

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Hyperbolic Functions

$\sinh x = \frac{e^x - e^{-x}}{2}, \qquad \cosh x = \frac{e^x + e^{-x}}{2}, \qquad \tanh x = \frac{\sinh x}{\cosh x}$

A cable hanging under its own weight takes the shape $y = c\cosh(x/c)$, which is why the catenary in Section 11.3 uses these functions. Example: $\cosh 1 = (2.71828 + 0.36788)/2 = 1.5431$.

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Sequences, Infinite Series and Power Series

An infinite series $\sum a_n$ converges if its partial sums approach a finite limit. For civil engineering problems, the key tests are:

• Divergence test. If $a_n \not\to 0$, the series diverges.
• Geometric series. $\sum a r^{n}$ converges to $a/(1-r)$ when $\lvert r \rvert < 1$.
• p-series. $\sum 1/n^p$ converges only when $p > 1$. The harmonic series ($p = 1$) diverges.
• Ratio test. If $L = \lim \lvert a_{n+1}/a_n \rvert < 1$, the series converges; if $L > 1$, it diverges.

A power series $\sum c_n (x - a)^n$ converges within a radius of convergence $R$ found from the ratio test. Taylor series about $x = a$ (Maclaurin when $a = 0$):

$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n$

These series justify everyday engineering approximations, such as $\sin\theta \approx \theta$, $\cos\theta \approx 1 - \theta^2/2$, and the slope correction $h^2/(2s)$ in taping.

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Partial Derivatives and the Total Differential

For $z = f(x, y)$, the partial derivative $\partial z/\partial x$ treats $y$ as constant. The total differential estimates how $z$ changes when both variables change slightly:

$dz = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y}dy$

Error example. A rectangular footing is measured as $2.00 \pm 0.01\text{ m}$ by $3.00 \pm 0.01\text{ m}$. Area $A = xy = 6.00\text{ m}^2$, and $dA = y\,dx + x\,dy = 3(0.01) + 2(0.01) = 0.05\text{ m}^2$, or about $0.8\%$.

Second partials give curvature, and setting $\partial f/\partial x = \partial f/\partial y = 0$ locates critical points of a two-variable function. A critical point is a minimum when $f_{xx} > 0$ and $f_{xx}f_{yy} - f_{xy}^2 > 0$.

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Applications of Integration: Force and Work

• Work by a variable force: $W = \int_a^b F(x)\,dx$. For a spring, $W = \frac{1}{2}k(x_2^2 - x_1^2)$.
• Pumping liquid: a horizontal slice at depth $y$ has weight $\gamma A(y)\,dy$ and must be lifted a distance $L(y)$, so $W = \int \gamma A(y) L(y)\,dy$.
• Hydrostatic force on a vertical plate: $F = \int \gamma\, y\, w(y)\,dy$, where $w(y)$ is the plate width at depth $y$.

Example. A cylindrical tank $2\text{ m}$ in radius and $5\text{ m}$ deep is full of water. All of it is pumped over the rim. Each slice of thickness $dy$ at depth $y$ below the rim weighs $9.81\pi(2)^2\,dy$ kN and is lifted $y$ meters:

$W = \int_0^5 9.81(4\pi)\,y\,dy = 9.81(4\pi)\frac{25}{2} = 1{,}541\text{ kJ}$

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Double and Triple Integrals

A double integral $\iint_R f(x,y)\,dA$ sums $f$ over a plane region. With $f = 1$ it gives area; with $f$ as a height it gives volume. Triple integrals $\iiint f\,dV$ give volume, mass and moments of solids.

Example. The volume under $z = 6 - x - 2y$ over the rectangle $0 \le x \le 2$, $0 \le y \le 1$:

$V = \int_0^1 \int_0^2 (6 - x - 2y)\,dx\,dy = \int_0^1 (12 - 2 - 4y)\,dy = 10 - 2 = 8$

In polar coordinates, $dA = r\,dr\,d\theta$. That form makes circular regions simple: the area of a circle is $\int_0^{2\pi}\int_0^R r\,dr\,d\theta = \pi R^2$.

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Systems of Linear Differential Equations

Coupled first-order equations such as $x' = a x + b y$ and $y' = c x + d y$ describe two interacting quantities. Examples are two connected tanks or two masses on springs. Write the system as $\mathbf{X}' = A\mathbf{X}$. For each eigenvalue $\lambda$ of $A$ with eigenvector $\mathbf{v}$, $e^{\lambda t}\mathbf{v}$ is a solution, and the general solution is their combination.

Example. $x' = y$ and $y' = 4x$. The eigenvalues satisfy $\lambda^2 - 4 = 0$, so $\lambda = \pm 2$. Hence $x = C_1 e^{2t} + C_2 e^{-2t}$, and $y = x' = 2C_1 e^{2t} - 2C_2 e^{-2t}$. Elimination gives the same result: differentiate the first equation to get $x'' = 4x$.

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Laplace Transforms

$\mathcal{L}\{f(t)\} = F(s) = \int_0^\infty e^{-st} f(t)\,dt$

Solving an initial-value problem. For $y' + 2y = 4$ with $y(0) = 0$, transforming term by term gives $sY + 2Y = 4/s$. So $Y = \dfrac{4}{s(s+2)} = \dfrac{2}{s} - \dfrac{2}{s+2}$ by partial fractions, and the inverse transform is $y = 2 - 2e^{-2t}$.