Math Problem Statement

Evaluate the surface integral ∬_S (x + y + z) dS, where S is the parallelogram with parametric equations x = u + v, y = u - v, z = 1 + 2u + v, 0 ≤ u ≤ 2, 0 ≤ v ≤ 1.

Solution

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Math Problem Analysis

Mathematical Concepts

Surface integrals
Vector calculus
Parametric surfaces

Formulas

Surface integral: ∬_S f(x, y, z) dS = ∬_R f(r(u, v)) ||r_u × r_v|| dudv
Cross product: r_u × r_v

Theorems

Surface integral computation in parametric form

Suitable Grade Level

Undergraduate (Calculus III or Vector Calculus)

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