Reference no: EM131003760

Suppose that f(x, y) is defined over a domain Ω ⊂ R² with boundary defined by a curve Γ. The normal derivative is the directional derivative in the direction of the outward-pointing normal vector to Γ and is denoted by ∂f / ∂n; that is, D_nf. The tangential derivative is the directional derivative in the direction of the unit tangent vector T to Γ, D_Tf. If we consider a vector function r(t) that traces Γ, then the unit tangent vector is given by T(t) = r'(t)/||r'(t)|| while the unit normal vector is given by n(t) = T'(t)/||T'(t)||.

Suppose that f(x, y) = x²y⁴ is defined over a domain Ω = {{x, y)|x² + y² ≤ 1} (a circle of radius one in the xy-plane). The boundary Γ is then traced by r(t) = (cos(t), sin(t)). At the point P (√2 / 2, √2 / 2) (corresponding to t = Π/4), the unit tangent vector is T = (-√2 / 2, √2 / 2)while the (outward) unit normal vector is n = (√2 / 2, √2 / 2).

a. Calculate the normal derivative of f(x, y) at P.

b. Calculate the tangential derivative of f(x, y) at P.