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Calculus 254.01
Formulas
44
Mathematics
Undergraduate 2
12/02/2011

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Cards

Term
Partial derivative of ƒ(x,y) with respect to x
Definition

ƒx(x,y) = limh→0 [ƒ(x+h,y)-ƒ(x,y)]/h

 

Term
Equation of tangent plane to surface z = ƒ(x,y) at the point P(x0,y0,z0)
Definition
z-z0 = ƒx(x0,y0)(x-x0) + ƒy(x0,y0)(y-y0)
Term
Linear approximation
Definition
L(x,y) = ƒ(a,b) + ƒx(a,b)(x-a) + ƒy(a,b)(y-b)
Term
Differential dz
Definition
dz = (∂z/∂x)dx + (∂z/∂y)dy
Term
Differential of z = (g(t),h(t))
Definition
(dz/dt) = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)
Term
Implicit differentiation solution for (dy/dx)
Definition
(dy/dx) = -(∂F/∂x)/(∂F/∂y) = -(Fx)/(Fy)
Term
Directional derivative of ƒ at (x0,y0) in the direction of a unit vector u = <a,b>
Definition
Duƒ(x0,y0) = ƒx(x,y)a + ƒy(x,y)b
Term
Gradient vector function of ƒ
Definition

∇ƒ(x,y)

=

xyz>

=

(∂ƒ/∂x)i + (∂ƒ/∂y)j + (∂ƒ/∂z)k

Term
Tangent plane to level surface F(x,y,z) = k at P(x0,y0,z0)
Definition
Fx(x0,y0,z0)(x-x0) + Fy(x0,y0,z0)(y-y0) + Fz(x0,y0,z0)(z-z0) = 0
Term
Normal line passing through P(x0,y0,z0)
Definition
(x-x0)/(Fx(x0,y0,z0)) = (y-y0)/(Fy(x0,y0,z0)) = (z-z0)/(Fz(x0,y0,z0))
Term
Local maximum, minimum, saddle point where both ƒx(a,b) = 0 and ƒy(a,b) = 0
Definition

D = ƒxx(a,b)ƒyy(a,b) - [ƒxy(a,b)]2

 

Minimum if D > 0 and ƒxx(a,b) > 0

Maximum if D > 0 and  ƒxx(a,b) < 0

Saddle Point if D < 0

Term
Langrange multiplier
Definition

∇ƒ(x0,y0,z0) = λ∇g(x0,y0,z0)

 

where ∇g(x0,y0,z0) = k

Term
Volume of solid that lies above rectangle R and below surface z = ƒ(x,y)
Definition
V = ∫∫R ƒ(x,y)dA
Term
Average value of a function of two variables defined on a rectangle R
Definition
ƒave = (1/A(R))∫∫R ƒ(x,y)dA
Term
Integral of z(x,y) over region D
Definition

∫∫ƒ(x,y) dA

=

abg1(x)g2(x) ƒ(x,y) dy dx

=

cdh1(y)h2(y) ƒ(x,y) dx dy

Term
Change to polar coordinates in a double integral
Definition
∫∫D ƒ(x,y) dA = ∫αβh1(θ)h2(θ) ƒ(rcosθ,rsinθ)r dr dθ
Term
Mass of lamina
Definition
m = ∫∫D ρ(x,y) dA
Term
Moment of lamina about axes
Definition

Mx = ∫∫D yρ(x,y) dA

 

My = ∫∫D xρ(x,y) dA

Term
Coordinates (x,y) of the center of mass of a lamina
Definition

x = (1/m)∫∫D xρ(x,y) dA

 

y = (1/m)∫∫D yρ(x,y) dA

Term
Triple integral
Definition

∫∫∫E ƒ(x,y,z) dV = ∫∫D[∫u1(x,y)u2(x,y)ƒ(x,y,z) dz] dA

= ∫abg1(x)g2(x)u1(x,y)u2(x,y)ƒ(x,y,z) dz dy dx...

Term
Mass of object E
Definition
m = ∫∫∫E ρ(x,y,z) dV
Term
Triple integral in cylindrical coordinates
Definition

∫∫∫E ƒ(x,y,z) dV

=

αβh1(θ)h2(θ)u1(rcosθ,rsinθ)u2(rcosθ,rsinθ)ƒ(rcosθ,rsinθ,z)r dz dr dθ

Term
Conversions from spherical to rectangular coordinates
Definition

x = ρsinΦcosθ

y = ρsinΦsinθ

z = ρcosΦ

ρ= x2 + y2 + z2

Term
Triple integral in spherical coordinates
Definition

∫∫∫E ƒ(x,y,z) dV

=

cdαβab ƒ(ρsinΦcosθ,ρsinΦsinθ,ρcosΦ)ρ2sinΦ dρ dθ dΦ

Term
Jacobian of transformation T given by x = g(u,v) and y = (u,v)
Definition
[∂(x,y)/∂(u,v)] = (∂x/∂u)(∂y/∂v) - (∂x/∂v)(∂y/∂u)
Term
Change of variable in a double integral
Definition

∫∫R ƒ(x,y) dA

=

∫∫S ƒ(x(u,v),y(u,v)) /∂(x,y)/∂(u,v)/ du dv

Term
Equation for a line integral
Definition

C ƒ(x,y) ds

=

ab ƒ(x(t),y(t)) √(dx/dt)2 + (dy/dt)2 dt

or for 3D

ab ƒ(x(t),y(t),z(t)) √(dx/dt)2 + (dy/dt)2 + (dz/dt)2 dt 

 

 

 

Term
Line integral with respect to arc length
Definition

C ƒ(x,y) dx = ∫ab ƒ(x(t),y(t)) x'(t) dt

 

C ƒ(x,y) dy = ∫ab ƒ(x(t),y(t)) y'(t) dt

Term
Parametric representation of a line segement that begins at r0 and ends at r1
Definition

r(t) = (1-t)r0 +tr1

 

0 < t < 1

Term
Line integral for a conservative vector field
Definition
C ∇ƒ•dr = ƒ(r(b)) - ƒ(r(a))
Term
Conditions for conservative vector field
Definition
(∂P/∂y) = (∂Q/∂x)
Term
Green's Theorem
Definition
C P dx + Q dy = ∫∫D ((∂Q/∂x) - (∂P/∂y)) dA
Term
curl F
Definition

∇ X F

=

[(∂R/∂y)-(∂Q/∂z)]i + [(∂P/∂z)-(∂R/∂x)]j + [(∂Q/∂x)-(∂P/∂y)]k

Term
div F
Definition
∇•F = (∂P/∂x) + (∂Q/∂y) + (∂R/∂z)
Term
Parametric surface
Definition
r(u,v) - x(u,v)i + y(u,v)j + z(u,v)k
Term
Tangent plane to parametric surface
Definition

rv = (∂x/∂v)(u0,v0)i + (∂y/∂v)(u0,v0)j + (∂z/∂v)(u0,v0)k


ru = (∂x/∂u)(u0,v0)i + (∂y/∂u)(u0,v0)j + (∂z/∂u)(u0,v0)k


with normal vector ru X rv

Term
Surface area
Definition

A(S) = ∫∫D /ru X rv/ dA

 

where ru = (∂x/∂u)i + (∂y/∂u)j + (∂z/∂u)k


and rv = (∂x/∂v)i + (∂y/∂v)j + (∂z/∂v)k

Term
Surface area where z = ƒ(x,y)
Definition
A(S) = ∫∫D √1 + (∂z/∂x)2 + (∂z/∂y)2 dA
Term
Surface integral
Definition
∫∫S ƒ(x,y,z) dS = ∫∫D ƒ(r(u,v)) /ru X rv/ dA
Term
Surface integral where z = g(x,y)
Definition

∫∫S ƒ(x,y,z) dS

=

∫∫D ƒ(x,y,g(x,y)) √(∂z/∂x)2 + (∂z/∂y)2 + 1 dA

Term
Surface integral of vector field
Definition
∫∫S F•dS = ∫∫D F•(ru X rv) dA
Term
Surface integral of vector field where z = g(x,y)
Definition
∫∫S F•dS = ∫∫D (-P(∂g/∂x) - Q(∂g/∂y) + R) dA
Term
Stokes Theorem
Definition
C F•dr = ∫∫S curlF•dS
Term
Divergence Theorem
Definition
∫∫S F•dS = ∫∫∫E divF dV
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