Calculus III Formula Sheet Math 2210, Spring 2010
• Resultant of two forces: ||F
+ F2
2
2
|| = ||F || + ||F ||
2
|| ||||F || cos φ • For the angle α between F and the resultant force: sin α = ||F||F+ ||F || sin φ × v|| for 0 ≤ θ ≤ 90 • Angle θ between vectors: cos θ = ||uu||||· vv|| , or sin θ = ||||uu|||| v|| v ·b • Projection Projection of v onto b: proj v = ||b|| b • Work from P to Q with force F: W = F · P−→Q = ||F||||P−→Q|| cos θ • Area of a parallelogram A = ||u × v|| • Volume of a parallelepiped V = |u · (v × w)| −→ −→ • Torque or scalar moment of F at P : P : T = ||P Q × F|| = ||P Q||||F|| sin θ |ax √ + by + cz + d| • Distance between a point P ( P (x , y , z ) and a plane D = a +b +c x = r cos θ r = √ x + y • Cylindrical coordinates: y = r sin θ , tan θ = y/x z = z z = z x = ρ sin φ cos θ ρ = √ x + y + z • Spherical coordinates: y = ρ sin φ sin θ , tan θ = y/x√ cos φ = z/ x + y + z z = ρ cos φ 1
1
2
+ 2 F1
2
2
1
1
b
◦
2
0
0
0
0
0
0
2
2
2
2
2
2
2
2
2
b
2
2
t
2
• Arc length L = ||r (t)|| dt, ||r (u)|| du dt, arc length parametrization s =
a
t0
r (t)
T (t)
• Unit tangent T(t) = ||r (t)|| ; Unit normal: N(t) = ||T (t)||
• Curvature κ(s) = ||r (s)||, κ(t) = ||||Tr ((tt))|||| = ||r (||tr) ×(t)r|| (t)|| • Osculating circle passing through a point r(t ) has radius r = 1/κ 1 /κ((t ). • Level curves for z = f ( f (x, y): f ( f (x, y) = k, where k is a real number in the range of f
3
0
• Implicit derivative of F ( F (x, y) = 0, where y depends on x:
0
dy = dx
− ∂F/∂x ∂F/∂y
1
2
• Projectile motion: r(t) = (v t cos α) i + (s + v t sin α − gt ) j. Speed: ||r (t)|| • Total differential of z = f (x, y): dz = f (x, y)dx + f (x, y)dy • Local linear approximation: L(x, y) = f (x , y ) + f (x , y )(x − x ) + f (x , y )(y − y ) • The tangent plane to f at (x , y ) is the LLA given by L • The formula for L above also implies that ∆z = f (x, y) − f (x , y ) ≈ f (x , y )∆x + 0
0
0
x
y
0
0
2
x
0
0
0
y
0
0
0
0
0
0
x
0
0
0
f y (x0, y0 )∆y.
• Directional derivative of f in the (unit) direction u: D f (x, y) = ∇f (x, y) · u. Here, u may be expressed as the vector cos φ, sin φ for some φ. • The normal vector n to a surface F (x,y,z ) = 0 at (x , y , z ) is n = ∇F (x , y , z ); note that if z = f (x, y), then F (x,y,z ) = f (x, y) − z may be used to get n. ∂ r ∂ r • The surface area over R of r = x(u, v)i + y(u, v) j + z (u, v)k is S = ∂u × ∂v dA u
0
0
0
0
0
0
R
• The surface area over R of z = f (x, y) is S = (z ) + (z ) + 1 dA • The x-center of mass of a region R is x¯ = M /M , where M = xδ (x, y) dA and M = δ (x, y) dA. The same formula carries over to 3D solids: M → M , A → V . dx dy dz • f (x, y(, z )) ds = f (x(t), y(t)(, z (t))) dt + dt + dt dt • Work W = · d , where = x + y + z ∂g ∂f • Green’s Theorem: · d = f (x, y) dx + g(x, y) dy = − dA ∂x ∂y ∂ ∂ • f (x,y,z ) dS = f (x(u, v), y(u, v), z (u, v)) ∂u × ∂v dA = f (x,y,g(x, y)) (z ) + (z ) + 1 dA ∂ ∂ • Φ= · dS = f (x,y,z ) dS = · ∂u × ∂v dA = · ∇G dA • Divergence Theorem: · dS = div dV • Stokes’ Theorem: · ds = · d = (curl ) · dS , = f + g + h x
R
2
y
y
2
y
R
y
R
2
b
C
yz
2
2
a
F
r
r
i
j
k
C
F
r
C
C
R
r
σ
r
R
x
R
F n
r
F
σ
σ
F n
y
2
F
R
R
F
σ
G
F T
C
r
2
F
C
r
F
σ
n
F
i
j
k