Multi Variable Calculus

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.

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 ...