Coordinate Systems

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 $(-2,3)$? ...

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.

Curvilinear Coordinates Exercise 1 : Inverting Spherical Coordinates Invert the transformation $x = r\sin\theta\cos\phi$, $y = r\sin\theta\sin\phi$, $z = r\cos\theta$ to express $(r, \theta, \phi)$ in terms of $(x, y, z)$. Exercise 2 : Coordinate Examples (Conversions) Invert the transformation $x = r\sin\theta\cos\phi$, $y = r\sin\theta\sin\phi$, $z = r\cos\theta$ to express $(r, \theta, \phi)$ in terms of $(x, y, z)$. Express the point $(x, y, z) = (2, -2, 1)$ in spherical and cylindrical coordinates. Express $(r, \theta, \phi) = (3, \pi/3, \pi/4)$ in Cartesian coordinates. Express $(s, \phi, z) = (4, 5\pi/6, -2)$ in Cartesian coordinates. Exercise 3 : Spherical Unit Vectors Express the spherical unit vectors $\hat{\mathbf{r}}$, $\hat{\boldsymbol{\theta}}$, $\hat{\boldsymbol{\phi}}$ in terms of the Cartesian unit vectors $\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$, $\hat{\mathbf{z}}$, i.e., derive: ...