Math Problem Statement

Show that F = (e^x cos y + yz)i + (xz - e^x sin y)j + (xy + z)k is conservative over its natural domain, and find a potential function for it.

Solution

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Math Problem Analysis

Mathematical Concepts

Vector Calculus
Conservative Vector Fields
Gradient Theorem

Formulas

∂P/∂y = ∂N/∂z
∂N/∂x = ∂M/∂z
∂M/∂y = ∂P/∂x
Potential function f(x, y, z): ∇f = F

Theorems

Theorem: A vector field is conservative if it is the gradient of some scalar function.

Suitable Grade Level

Undergraduate Mathematics

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