Math 250A Formula Sheet Alternate Form of Dot Product Ð ⇀ ⇀ ⇀ ⇀ u ⋅Ð v = ∣∣Ð u ∣∣∣∣Ð v ∣∣ cos θ Total Differential dz = fx (x, y)dx + fy (x, y)dy ⇀ ⇀ Projection of Ð u onto Ð v Ð ⇀ Ð ⇀ u ⋅ v Ð ⇀ ⇀ ⇀ ProjÐ )Ð v v u =( Ð ∣∣ ⇀ v ∣∣2 Chain Rule dw ∂w dx ∂w dy = + dt ∂x dt ∂y dt Property of Cross Product ⇀ ⇀ ⇀ ⇀ ∣∣Ð u ×Ð v ∣∣ = ∣∣Ð u ∣∣∣∣Ð v ∣∣ sin θ Implicit Differentiation dy Fx (x, y) =− dx Fy (x, y) ⇀ ⇀ Note: ∣∣Ð u ×Ð v ∣∣ gives the area of a parallelogram ⇀ ⇀ determined by Ð u and Ð u Cylindrical to Rectangular x = r cos θ y = r sin θ r 2 = x2 + y 2 x tan θ = y Spherical to Rectangular x = ρ sin φ cos θ z = ρ cos φ y = ρ sin φ sin θ Unit Ð ⇀ T= ρ 2 = x2 + y 2 + z 2 Tangent Vector Ð ⇀ Ð ⇀ r ′ (t) v (t) = Ð ⇀ Ð ∣∣ r ′ (t)∣∣ ∣∣ ⇀ v (t)∣∣ Principal Unit Normal Vector Ð ⇀ Ð ⇀ ⇀ ⇀ Ð ⇀ T ′ (t) v ×Ð a ×Ð v N= Ð = Ð ⇀ Ð ⇀ Ð ⇀ ⇀′ ∣∣ T (t)∣∣ ∣∣ v × a × v ∣∣ Tangential and Normal Comp. of Acceleration ⇀ Ð ⇀ Ð ⇀ Ð ⇀ = Dt [∣∣ v ∣∣] = a ⋅ T aÐ T ⇀ ⇀ ⇀ Ð ⇀ ∣∣Ð v ×Ð a ∣∣ Ð ⇀ Ð ⇀ Ð ⇀ = ∣∣ v ∣∣∣ T ′ ∣∣∣ = = aÐ a ⋅ N = Ð ⇀ N ∣∣ v ∣∣ √ ⇀ ⇀2 ∣∣Ð a ∣∣2 − aÐ T Directional Derivative Du f (x, y) = fx (x, y) cos θ + fy (x, y) sin θ ⇀ or if Ð u is a unit vector: Ð ⇀ ⇀ D f (x, y) = ∇ f (x, y) ⋅ Ð u u Tangent Plane to a Surface Fx (x0 , y0 , z0 )(x − x0 ) + Fy (x0 , y0 , z0 )(y − y0 ) + Fz (x0 , y0 , z0 )(z − z0 ) = 0 Second Partials Test d = fxx (a, b)fyy (a, b) − [fxy (a, b)]2 First Moments and Center of Mass m = ∬ ρ dA, Mx = ∬ yρ dA, My = ∬ xρ dA R R R My Mx (¯ x, y¯) = ( , ) m m Second Moments and Polar Moment of Inertia Ix = ∬ y 2 ρ dA, Iy = ∬ x2 ρ dA R R I0 = Ix + Iy Surface Area √ S=∬ 1 + fx 2 + fy 2 dA R Arc Length s=∫ b a ⇀ ∣∣Ð r ′ (t)∣∣ dt Curvature Ð ⇀ ⇀ ⇀ ∣∣ T ′ (t)∣∣ ∣∣Ð v ×Ð a ∣∣ ∣y ′′ ∣ K= Ð = = ⇀ Ð ⇀ [1 + (y ′ )2 ]3/2 ∣∣ r ′ (t)∣∣ ∣∣ v ∣∣3 Change of Variables: Spherical ∭ f (x, y, z) dV = Q ∫ θ2 θ1 ∫ φ2 φ1 ∫ ρ2 g(ρ, θ, φ)ρ2 sin φ dρ dφ dθ ρ1 where g is the function f converted to spherical Jacobian ∂(x, y) ∂x ∂y ∂y ∂x = − ∂(u, v) ∂u ∂v ∂u ∂v Line Integral ∫ f (x, y) ds = ∫ C b a ⇀ f (x(t), y(t))∣∣Ð r ′ (t)∣∣ dt Line Integral of a Vector Field Ð ⇀ Ð ⇀ ∫ F ⋅dr =∫ C bÐ ⇀ a ⇀ F ⋅Ð r ′ (t) dt if conservative: f (x(b), y(b)) − f (x(a), y(a)) Green’s Theorem ∫ M dx + N dy = ∬ ∂N ∂x − ∂M ∂y dA R C Surface Integral √ ∬ f (x, y, z) dS = ∬ f (x, y, g(x, y)) 1 + gx 2 + gy 2 dA S R Upward Oriented Flux Ð ⇀ Ð ⇀ Ð ⇀ ∬ F ⋅ N dS = ∬ F ⋅ ⟨−gx , −gy , 1⟩ dA R S Divergence Theorem Ð ⇀ Ð ⇀ Ð ⇀ ∬ F ⋅ N dS = ∭ div( F ) dV S Q Stokes’ Theorem Ð ⇀ Ð Ð ⇀ Ð ⇀ ⇀ ∫ F ⋅ d r = ∬ curl( F ) ⋅ N dS C S 2
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