Advanced Calculus Formulas a = ha1 , a2 , a3 i r = hx, y, zi

b = hb1 , b2 , b3 i r0 = hx0 , y0 , z0 i

c = hc1 , c2 , c3 i n = hn1 , n2 , n3 i

Dot (Scalar) Product: a · b = a1 b1 + a2 b2 + a3 b3 = |a| |b| cos θ p √ Magnitude: |a| = a21 + a22 + a23 = a · a Projections:

proja b =

a·b a a·a i j k a × b = a1 a2 a3 b b b 1 2 3

Cross (Vector) Product:

|a × b| = |a| |b| sin θ Box (Triple Scalar) Product:

a1 a2 a3 (a × b) · c = b1 b2 b3 c c c 1 2 3

Distance Formulas Two Points:

r1 and r2

Point and a Line:

d = |r2 − r1 |

r1 and r0 + t a d =

r1 and (r − r0 ) · n = 0

Point and a Plane:

d = Two Parallel Planes:

|D2 − D1 | |n|

r1 + t a1 and r2 + t a2 d =

Two Parallel Lines:

|(r1 − r0 ) · n| |n|

r · n = D1 and r · n = D2 d =

Two Skew Lines:

|(r1 − r0 ) × a| |a|

|(r2 − r1 ) · (a1 × a2 )| |a1 × a2 |

r1 + t a and r2 + t a d =

|(r2 − r1 ) × a| |a|

Lines and Planes Lines in Space: parametric representation r(t) = r0 + t a non-parametric representation y − y0 z − z0 x − x0 = = a1 a2 a3 Planes in Space: parametric representation r(s, t) = r0 + s a + t b non-parametric representation (r − r0 ) · n = 0 Tangent Lines: explicitly defined curves:

y = f (x)

y − y0 = f 0 (x0 ) (x − x0 ) implicity defined (level) curves:

f (x, y) = c

fx (x0 , y0 ) (x − x0 ) + fy (x0 , y0 ) (y − y0 ) = 0 or in vector form,

(r − r0 ) · ∇f (x0 , y0 ) = 0

Tangent Planes: explicitly defined surfaces:

z = f (x, y)

z = f (x0 , y0 ) + fx (x0 , y0 ) (x − x0 ) + fy (x0 , y0 ) (y − y0 ) implicity defined (level) surfaces:

F (x, y, z) = c

Fx (x0 , y0 , z0 ) (x − x0 ) + Fy (x0 , y0 , z0 ) (y − y0 ) + Fz (x0 , y0 , z0 ) (z − z0 ) = 0 or in vector form,

(r − r0 ) · ∇F (x0 , y0 , z0 ) = 0

Derivatives f = hf1 , . . . , fm i ,

x = hx1 , . . . , xn i ,

t = ht1 , . . . , tk i

Matrix of Partial Derivatives:    ∂ (f1 , . . . , fm ) =  Dx f =  ∂ (x1 , . . . , xn ) 

Gradient:

 ∇=

∂f1 ∂ x1

· · ·

· · ·

· · ·

∂fm ∂ x1

∂ ∂ , ... , ∂ x1 ∂ xn

∂f1 ∂ xn

· · ·

∂fm ∂ xn

     



Chain Rule: F (t) = f (x(t)) ∂Fi ∂fi ∂ x1 ∂fi ∂ xn = + ··· + ∂ tj ∂ x1 ∂ tj ∂ xn ∂ tj for each i = 1, . . . , m and j = 1, . . . , k

Product Rules: If x(t) and y(t) are differentiable vector-valued functions, then dx dy d (x · y) = ·y+x· dt dt dt d dx dy (x × y) = ×y+x× dt dt dt Directional Derivatives: derivative of f (x) in the direction of v Dv f (x) = ∇f (x) ·

v |v|

Leibniz’s Rule: Z b(t) Z b(t) d d f (x, t) dx = f (x, t) dx + b0 (t) f (b(t), t) − a0 (t) f (a(t), t) dt a(t) a(t) dt

Parametric Curves in