8.3 From Test 3

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Solution.

(a)

Write \(w = x+2y+3z\). The awkward parts of the three components are \(\dfrac 1w\), \(\dfrac 2w\), \(\dfrac 3w\), which are exactly \(\nabla \ln w\), since \(w_x = 1\), \(w_y = 2\), \(w_z = 3\). The remaining parts \(-3x\), \(y^2\), \(-1\) each involve only their own variable. Hence \[f = \ln \left |x+2y+3z\right | - \frac {3x^2}{2} + \frac {y^3}{3} - z\] satisfies \(\nabla f = \textbf {F}\), as differentiating confirms, so the field is conservative.

(b)

\[\int _C\textbf {F}\cdot d\overline {r} = f(2,3,4)-f(1,0,0).\] At \((2,3,4)\): \(w = 2+6+12 = 20\), giving \(\ln 20 - 6 + 9 - 4 = \ln 20 - 1\). At \((1,0,0)\): \(w=1\), giving \(0 - \dfrac 32 + 0 - 0 = -\dfrac 32\). Hence \[\int _C\textbf {F}\cdot d\overline {r} = \ln 20 - 1 + \frac 32 = \frac 12 + \ln 20 \approx 3.496 .\]

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Solution. Complete the square in the denominator: \(s^2-4s+13 = (s-2)^2+9\). Then write the numerator in terms of \(s-2\): \[s+4 = (s-2)+6 ,\] so \[F(s) = \frac {s-2}{(s-2)^2+9} + \frac {6}{(s-2)^2+9}.\] By the first shifting theorem, \[f(t) = e^{2t}\Big (\cos 3t + 2\sin 3t\Big ),\] the \(2\) arising as \(\dfrac 63\) because the sine transform carries the \(3\) in its numerator.

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Solution. The function is \(3t-t^2\) switched on at \(t=0\) and off at \(t=3\): \[f(t) = \big (3t-t^2\big )\Big [u(t)-u(t-3)\Big ] = \big (3t-t^2\big ) - \big (3t-t^2\big )u(t-3).\] For the second term the second shifting theorem needs the function written in \(\tau = t-3\). With \(t = \tau +3\), \[3t-t^2 = 3(\tau +3)-(\tau +3)^2 = -\tau ^2-3\tau ,\] so \[\mathcal {L}\big \{(3t-t^2)u(t-3)\big \} = e^{-3s}\Big (-\frac {2}{s^3} - \frac {3}{s^2}\Big ).\] Hence \[\mathcal {L}\{f\} = \frac {3}{s^2} - \frac {2}{s^3} + e^{-3s}\Big (\frac {2}{s^3} + \frac {3}{s^2}\Big ).\]

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Solution. The left side is \(y * \cos t\), so transforming turns it into a product: \[Y(s)\cdot \frac {s}{s^2+1} = \frac {1}{s^2} - \frac {2}{s^3}.\] Hence \[Y(s) = \frac {s^2+1}{s}\Big (\frac {1}{s^2}-\frac {2}{s^3}\Big ) = \frac {s^2+1}{s^3} - \frac {2\big (s^2+1\big )}{s^4} = \frac 1s + \frac {1}{s^3} - \frac {2}{s^2} - \frac {2}{s^4}.\] Inverting term by term, \[y(t) = 1 - 2t + \frac {t^2}{2} - \frac {t^3}{3}.\]

Course Outline

Rationale

Multivariable functions arise in many real world situations, where physical quantities often depend on two or more variables. This course takes calculus from the two dimensional world of single variable functions into the three dimensional world of multivariable functions which are required to understand and manipulate planes and surfaces, curves in two or three dimensions and scalar-valued and vector-valued functions of several variables which arise in many physical situations. The course also introduces Laplace transforms and Fourier series which constitute an important tool in solving problems that involve initial and boundary values problems. The course thus lays a foundation for application which arise in many physical model.

Objectives

At the end of this course, students should be able to :-

1.

Synthesise the key concepts of differential, integral and multivariable calculus.

2.

Use the standard techniques of multivariable calculus, both differential and integral, and utilise them to solve selected applied problems.

3.

Examine 3 dimensional coordinate systems and graph in 3 dimensions.

4.

Use double, triple, line and surface integrals in applications including Green’s theorem, Stokes theorem and divergence theorem.

5.

Find solutions to a variety of non-linear differential equations and systems of equations.

6.

Apply Laplace transform methods to solve differential equations.

7.

Use Fourier series and transforms and their properties to solve differential problems.

Pre-requisite: Analytic Geometry and Calculus

Course Content

1.

Functions of Several Variables Multivariable chain rule, implicit functions, determinants, functional dependence, transformations in 2 and 3 dimensions, implicit functions theorem and the inverse function theorem.

2.

Further Differential Calculus Directional derivatives and gradient vector; stationary points of functions of several variables, Extrema of functions of two variables and applications, Lagrange multipliers, Taylor’s theorem in several variables.

3.

Surfaces Quadric surfaces, Normal to a surface, tangent plane to a surface, tangent to a curve in
3 - D, rectification of a curve in 3 - D, parametric surfaces.

4.

Multiple Integration Double integrals over rectangles, Iterated integrals, Double integrals over general regions, change of variables: Jacobians, double integrals in polar coordinates. Triple integrals, integration in cylindrical and spherical coordinates.

5.

Vector Analysis Scalar and vector fields, the vector operators grad, div , curl and basic operations identities, Vector fields, line integrals, conservative vector fields and independence of path, Green’s theorem, Parametric surfaces, surface integrals, Stokes theorem, divergence theorem.

6.

Differential Equations Systems of differential equations, power series solution of first and second order equations at ordinary and regular singular points, stability of solutions of non linear differential equations.

7.

Laplace Transforms Definition and existence of Laplace transforms, properties of Laplace transforms, the inverse Laplace transform, Linearity, transforms of derivatives and integrals, \(s\)-Shifting, \(t-\)Shifting, the unit step function, Differentiation and integration of transforms, the convolution theorem, applications to initial value problems and systems of differential equations.

8.

Fourier series and integrals Inner products of functions, orthogonal functions, even and odd functions, Half-range expansions, approximation by trigonometric polynomials, complex form of Fourier series, Fourier integrals, Fourier Cosine and Sine transforms, Fourier transform.

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