Applied Mathematics

Transformation of Coordinates; Matrices Exercise 1 : Shift Paraboloid The vertex of the paraboloid shown in Fig. 14.18 is at a distance 2 from the origin of the coordinates. The equation is $z = 2 + x^2 + y^2$. What is the transformation which will shift the paraboloid so that its vertex coincides with the origin O? Exercise 2 : Line Under Translation The equation of a certain straight line is $y = -3x + 5$. What will its equation be in a new $x’$-$y’$ coordinate system due to a shift of the origin of $(-2,3)$? ...

Taylor Series and Power Series Exercise 1 : Expansion of a Function in a Power Series Expand the following functions at $x_0 = 0$ in a series up to the first four terms: $f(x) = \sqrt{1 - x}$ $f(t) = \sin(\omega t + \pi)$ $f(x) = \ln[(1 + x)^5]$ $f(x) = \cos x$ $f(x) = \cosh x$ Exercise 2 : Interval of Convergence of a Power Series Obtain the radius of convergence of the following series: ...

Surface Integrals, Divergence, Curl and Potential Exercise 1 : Coordinate Planes Given three squares with an area of 4 units each. They are placed: in the x-yplane, in the x-zplane, and in the y-zplane. Determine the surface elements. Exercise 2 : Rectangle Vector Element Given a rectangle with area $a \cdot b$, determine the vector element. Exercise 3 : Flux Through Given Surface Elements Compute the flow of the vector field $\mathbf{F}(x, y, z) = (5, 3, 0)$ through the surfaces given by the respective surface elements: ...

Multiple Integrals; Coordinate Systems Exercise 1 : Multiple Integrals with Constant Limits Evaluate the following multiple integrals: $\int_{y=0}^{b} \int_{x=0}^{a} dx dy$ $\int_{y=0}^{2} \int_{x=0}^{1} x^2 dx dy$ $\int_{x=0}^{\pi} \int_{y=0}^{\pi} \sin x \sin y dx dy$ $\int_{n=1}^{2} \int_{v=2}^{4} n(1+v) dv dn$ $\int_{x=-1/2}^{1/2} \int_{y=-1}^{1} \int_{z=0}^{2} dx dy dz$ $\int_{x=0}^{1} \int_{y=0}^{y_1} \int_{z=0}^{z_1} e^{az} dx dy dz$ Exercise 2 : Multiple Integrals with Variable Limits Evaluate the integrals: $\int_{x=0}^{2} \int_{y=x-1}^{3x} x^2 dy dx$ $\int_{x=0}^{1} \int_{y=0}^{2x} \int_{z=0}^{x+y} dx dy dz$ Using a double integral, obtain the area of an ellipse and the position of the center of mass of the half ellipse $(x \geq 0)$. The equation of an ellipse is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Exercise 3 : Coordinate Systems A point has Cartesian coordinates $P = (3, 3)$. What are its polar coordinates? Give the equation of a circle of radius $R$ in Cartesian coordinates and polar coordinates. Obtain the equation of the spiral shown in Fig. 13.22 in polar coordinates. Evaluate $\int_{\theta=0}^{\pi/4} \int_{r=0}^{a} r^2 \cos \theta , dr , d\theta$. Exercise 4 : Cylindrical Coordinates Compute the volume of the hollow cylinder shown in Fig. 13.23 using cylindrical coordinates. Evaluate the volume of a cone of radius $R$ and height $h$. Obtain the moment of inertia of the cone about its center axis. The density $\rho$ is a constant. Exercise 5 : Moment of Inertia Calculate the moment of inertia of a sphere of radius $R$ and of constant density $\rho$ about an axis through its center, using spherical coordinates.

Laplace Transforms Exercise 1 : Applying Laplace Transforms Obtain the Laplace transforms for the following functions: $\frac{1}{4}t^3$ $5e^{-2t}$ $4\cos 3t$ $\sin^2 t$ Exercise 2 : Inverse Laplace Transforms Obtain the inverse transforms for the following: $\frac{1}{4s^2 + 1}$ $\frac{1}{s(s + 4)}$ $\frac{2}{s(s^2 + 9)}$ $\frac{6}{1 - s^2}$ $\frac{1}{s^2(s^2 + 1)}$ $\frac{4}{s(s^2 - 6s + 8)}$ Exercise 3 : Solving Linear ODEs via Laplace Solve the following linear differential equations with constant coefficients: ...

Functions of Several Variables; Partial Differentiation; and Total Differentiation Exercise 1 : Table of Variables Construct a table of values for the function $f(x, y) = x^2 y + 6$ where $x = -2, -1, 0, 1$ and $y = -2, -1, 0, 1, 2$. Exercise 2 : Plane and Quadratic Surfaces What surfaces are represented by the following functions? Sketch them! $z = -x - 2y + 2$ $z = x^2 + y^2$ $z = \sqrt{1 - \frac{x^2}{4} - \frac{y^2}{9}}$ Exercise 3 : Partial Derivatives — Practice Obtain the partial derivatives of ...

Exponential, Logarithmic and Hyperbolic Functions Exercise 1 : Basic Exponent Rules Calculate the terms given in the next questions or give a transformation: $a^{-n}$ $27^{\frac{1}{3}}$ $a^{\frac{1}{n}}$ $(0,1)^0$ $(y^3)^2$ $x^{-\frac{3}{2}}$ $10^3 \cdot 10^{-3} \cdot 10^2$ $3^{-3}$ Exercise 2 : Roots and Powers Calculate the terms given in the next questions or give a transformation: $(\sqrt{2})^2$ $e^{\frac{1}{10}}$ $(\ln 2)^0$ $\sqrt{5} \cdot \sqrt{7}$ $(0,5)^2 \cdot (0,5)^{-4} \cdot (0,5)^0$ $\sqrt{8} \cdot \sqrt{3}$ Exercise 3 : Common Logarithms (base 10) Calculate the terms given in the next questions or give a transformation: ...

Eigenvalues and Eigenvectors of Real Matrices Exercise 1 : Eigenvalues — 2x2 real matrix For $A = \begin{pmatrix} 4 & 2 \ 1 & 3 \end{pmatrix}$ find the eigenvalues. Exercise 2 : Real vs Complex Eigenvalues — 2x2 question Is it possible for a real $2 \times 2$ matrix to have one real and one complex eigenvalue? Exercise 3 : No Real Eigenvalues — rotation-like matrix Prove that there are no real eigenvalues of the matrix $A = \begin{pmatrix} 3 & 2 \ -2 & 1 \end{pmatrix}$ ...

Differential Equations Exercise 1 : Classification — Linear vs Nonlinear Which of the following are linear first- and second-order DEs with constant coefficients? $y’ + x^2 y = 2x$ $5y’’ - 2y’ - 4x = 3y$ $y^4 + 2y’’ + 3y’ = 0$ $\sin x, y’’ - y = 0$ $y’’ - x^5 = 2$ $2y’’ - y’ + \frac{3}{2}y = 0$ Exercise 2 : Homogeneity and Order Which of the following are homogeneous and non-homogeneous DEs and what is the order in each case? ...

Differential Calculus Exercise 1 : Sequences and Limits of Sequences Calculate the limiting value of the following sequences for $n \to \infty$. $a_n = \frac{\sqrt{n}}{n}$ $a_n = \frac{5 + n}{2n}$ $a_n = \left(-\frac{1}{4}\right)^n - 1$ $a_n = \frac{2}{n} + 1$ $a_n = \frac{n^3 + 1}{2n^3 + n^2 + n}$ $a_n = 2 + 2^{-n}$ $a_n = \frac{n^2 - 1}{(n + 1)^2} + 5$ Exercise 2 : Limits of Functions Calculate the following limits: ...