Term
| Partial derivative of ƒ(x,y) with respect to x |
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Definition
ƒx(x,y) = limh→0 [ƒ(x+h,y)-ƒ(x,y)]/h
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Term
| Equation of tangent plane to surface z = ƒ(x,y) at the point P(x0,y0,z0) |
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Definition
| z-z0 = ƒx(x0,y0)(x-x0) + ƒy(x0,y0)(y-y0) |
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Term
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Definition
| L(x,y) = ƒ(a,b) + ƒx(a,b)(x-a) + ƒy(a,b)(y-b) |
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Term
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Definition
| dz = (∂z/∂x)dx + (∂z/∂y)dy |
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Term
| Differential of z = (g(t),h(t)) |
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Definition
| (dz/dt) = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt) |
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Term
| Implicit differentiation solution for (dy/dx) |
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Definition
| (dy/dx) = -(∂F/∂x)/(∂F/∂y) = -(Fx)/(Fy) |
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Term
| Directional derivative of ƒ at (x0,y0) in the direction of a unit vector u = <a,b> |
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Definition
| Duƒ(x0,y0) = ƒx(x,y)a + ƒy(x,y)b |
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Term
| Gradient vector function of ƒ |
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Definition
∇ƒ(x,y)
=
<ƒx,ƒy,ƒz>
=
(∂ƒ/∂x)i + (∂ƒ/∂y)j + (∂ƒ/∂z)k |
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Term
| Tangent plane to level surface F(x,y,z) = k at P(x0,y0,z0) |
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Definition
| Fx(x0,y0,z0)(x-x0) + Fy(x0,y0,z0)(y-y0) + Fz(x0,y0,z0)(z-z0) = 0 |
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Term
| Normal line passing through P(x0,y0,z0) |
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Definition
| (x-x0)/(Fx(x0,y0,z0)) = (y-y0)/(Fy(x0,y0,z0)) = (z-z0)/(Fz(x0,y0,z0)) |
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Term
| Local maximum, minimum, saddle point where both ƒx(a,b) = 0 and ƒy(a,b) = 0 |
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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 |
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Term
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Definition
∇ƒ(x0,y0,z0) = λ∇g(x0,y0,z0)
where ∇g(x0,y0,z0) = k |
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Term
| Volume of solid that lies above rectangle R and below surface z = ƒ(x,y) |
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Definition
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Term
| Average value of a function of two variables defined on a rectangle R |
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Definition
| ƒave = (1/A(R))∫∫R ƒ(x,y)dA |
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Term
| Integral of z(x,y) over region D |
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Definition
∫∫D ƒ(x,y) dA
=
∫ab∫g1(x)g2(x) ƒ(x,y) dy dx
=
∫cd∫h1(y)h2(y) ƒ(x,y) dx dy |
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Term
| Change to polar coordinates in a double integral |
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Definition
| ∫∫D ƒ(x,y) dA = ∫αβ∫h1(θ)h2(θ) ƒ(rcosθ,rsinθ)r dr dθ |
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Term
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Definition
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Term
| Moment of lamina about axes |
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Definition
Mx = ∫∫D yρ(x,y) dA
My = ∫∫D xρ(x,y) dA |
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Term
| Coordinates (x,y) of the center of mass of a lamina |
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Definition
x = (1/m)∫∫D xρ(x,y) dA
y = (1/m)∫∫D yρ(x,y) dA |
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Term
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Definition
∫∫∫E ƒ(x,y,z) dV = ∫∫D[∫u1(x,y)u2(x,y)ƒ(x,y,z) dz] dA
= ∫ab∫g1(x)g2(x)∫u1(x,y)u2(x,y)ƒ(x,y,z) dz dy dx... |
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Term
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Definition
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Term
| Triple integral in cylindrical coordinates |
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Definition
∫∫∫E ƒ(x,y,z) dV
=
∫αβ∫h1(θ)h2(θ)∫u1(rcosθ,rsinθ)u2(rcosθ,rsinθ)ƒ(rcosθ,rsinθ,z)r dz dr dθ |
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Term
| Conversions from spherical to rectangular coordinates |
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Definition
x = ρsinΦcosθ
y = ρsinΦsinθ
z = ρcosΦ
ρ2 = x2 + y2 + z2 |
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Term
| Triple integral in spherical coordinates |
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Definition
∫∫∫E ƒ(x,y,z) dV
=
∫cd∫αβ∫ab ƒ(ρsinΦcosθ,ρsinΦsinθ,ρcosΦ)ρ2sinΦ dρ dθ dΦ
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Term
| Jacobian of transformation T given by x = g(u,v) and y = (u,v) |
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Definition
| [∂(x,y)/∂(u,v)] = (∂x/∂u)(∂y/∂v) - (∂x/∂v)(∂y/∂u) |
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Term
| Change of variable in a double integral |
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Definition
∫∫R ƒ(x,y) dA
=
∫∫S ƒ(x(u,v),y(u,v)) /∂(x,y)/∂(u,v)/ du dv |
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Term
| Equation for a line integral |
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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
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Term
| Line integral with respect to arc length |
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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
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Term
| Parametric representation of a line segement that begins at r0 and ends at r1 |
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Definition
r(t) = (1-t)r0 +tr1
0 < t < 1 |
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Term
| Line integral for a conservative vector field |
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Definition
| ∫C ∇ƒ•dr = ƒ(r(b)) - ƒ(r(a)) |
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Term
| Conditions for conservative vector field |
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Definition
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Term
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Definition
| ∫C P dx + Q dy = ∫∫D ((∂Q/∂x) - (∂P/∂y)) dA |
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Term
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Definition
∇ X F
=
[(∂R/∂y)-(∂Q/∂z)]i + [(∂P/∂z)-(∂R/∂x)]j + [(∂Q/∂x)-(∂P/∂y)]k |
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Term
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Definition
| ∇•F = (∂P/∂x) + (∂Q/∂y) + (∂R/∂z) |
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Term
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Definition
| r(u,v) - x(u,v)i + y(u,v)j + z(u,v)k |
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Term
| Tangent plane to parametric surface |
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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
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Term
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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 |
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Term
| Surface area where z = ƒ(x,y) |
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Definition
| A(S) = ∫∫D √1 + (∂z/∂x)2 + (∂z/∂y)2 dA |
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Term
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Definition
| ∫∫S ƒ(x,y,z) dS = ∫∫D ƒ(r(u,v)) /ru X rv/ dA |
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Term
| Surface integral where z = g(x,y) |
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Definition
∫∫S ƒ(x,y,z) dS
=
∫∫D ƒ(x,y,g(x,y)) √(∂z/∂x)2 + (∂z/∂y)2 + 1 dA |
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Term
| Surface integral of vector field |
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Definition
| ∫∫S F•dS = ∫∫D F•(ru X rv) dA |
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Term
| Surface integral of vector field where z = g(x,y) |
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Definition
| ∫∫S F•dS = ∫∫D (-P(∂g/∂x) - Q(∂g/∂y) + R) dA |
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Term
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Definition
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Term
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Definition
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