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MA35 Labs

1 Single Variable Calculus

2 » Two Variable Calculus

3 Three Variable Calculus

Polar Coordinate Labs

Two Variable Functions

Contents

2.1 Functions of Two Variables

2.2 Differentiation

2.3 More Derivatives

2.4 Integration

2.4.1 Volume Under a Function Graph

2.4.2 Riemann Integral

2.4.3 Volume between Function Graphs

2.4.4 Change of Order of Integration

2.4.5 Surface Area for Function Graphs

2.4.6 Change of Variables

2.4.7 Total Mass of a Region in the Plane

2.4.8 Center of Mass

2.4.9 Moment of Inertia

2.5 Vector Analysis

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Change of Variables

Text

Integration in a non-Cartesian coordinate system requires an application of the Change of Variables Theorem.

Consider what would happen for a function of two variables. Up until now, we have integrated over two variables by adding up volumes of rectangular prisms with heights f(x, y) and rectangular bases of area dx * dy. If we decide to express x and y as functions of variables u and v, we need to find some new expression for the volume of the differential.

The height will still be the value of the function, expressed as f(x(u,v), y(u,v)).

The base of the volume will be a parallelogram. The two vectors representing the sides of this parallelogram are based on what happens when u and v change. When u changes by some amount du, x increases by x_u*du and y increases by y_u*du. The resulting vector is (x_u * du, y_u * du).

The other vector, which is based on what happens when v increases some amount dv, is (x_v * dv, y_v * dv).

To find the area of this base, take the cross product of the vectors, which is equal to

(x_uy_v - x_vy_u)dudv.

Finally, multiply by hieght to get the Change of Variables Theorem for double integrals:

∫∫_Df(x, y)dxdy = ∫∫_D*f(x(u, v), y(u, v))(x_uy_v - x_vy_u)dudv,

Where D is the domain in Cartesian coordinates and D* is the domain in the new coordinate system.

Note that the expression

(x_uy_v - x_vy_u)

is the determinant of the matrix with rows

(x_u, x_v), (y_u, y_v).

Demos

Change of Variables
	This demo has three windows. Two of these show the domains in Cartesian coordinates and in the new coordinate system. The third window shows a close-up of the parallelogram which is the base of an infinitesimal prism used in integration.

Exercises