Lecture 15 Multiple Integration
Lecture 15 Multiple Integration (Relevant section from Stewart, Section 15.1) We now turn to the integration of scalar-valued functions f : Rn R, i.e., f(x ,x , , c ), over → 1 2 · · · n regions in Rn. The need to perform such integrations is common in Physics. For example, we may wish to find: 1. The total charge Q in a region R R3 that is enclosed by a closed surface S, e.g., a spherical ⊂ surface of radius R centered at a point p R3, ∈ 2. The probability of finding an electron in the 1s state of a hydrogen atom within the distance α/a from the nucleus, where α 0 and a is the so-called Bohr radius. 0 ≥ 0 In all cases, the idea of integrating scalar-valued functions of several variables is a natural generalization of the integration of real-valued functions of a single variable, i.e., f : R R over the real line R. → We simply have to keep in mind the “Spirit of Calculus,” where we subdivide the region of interest and perform a summation over the subregions. Double Integrals (Integration in R2) (Relevant sections from Stewart, Section 15.1-2) We begin with the integration of functions of two variables, f(x,y), over regions in the plane R2. Suppose that a function f : R R2 is defined → over a region R R2. ⊂ Example: f(x,y) is the (2D or “areal”) density of a thin plate at (x,y), a measure of the amount of mass per unit area around the point (x,y).
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