Quadratic Surfaces

Multiple Integration

Average Value

Riemann Sum

Jacobians• Polar:

• Cylindrical:

• Spherical:

Mean Value Theorem

Center of Mass

𝑦 = 𝑚𝑥

𝑦 = 𝑚𝑥2

𝑦 = 𝑚 𝑥

• Rewrite as product of limits:

• Lines to Test

• Convert to Polar if

• Plug in values

• Squeeze Theorem

Limits

Limit Definitions

Equation of a Tangent Plane

Linear Approximation

Chain Rule for Paths

Chain Rule (Generalized)

Directional Derivatives

Multivariable Differentiation

Optimization

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Global Optimization

Second Derivative Test

Lagrange Multipliers

Vector-Valued Functions

Properties

Arc Length

Tangent Line Parametrization

Flux

Stokes’ Theorem

Surface Independence

• 𝑪𝒖𝒓𝒍 =𝑪𝒊𝒓𝒄𝒖𝒍𝒂𝒕𝒊𝒐𝒏

𝑼𝒏𝒊𝒕 𝑨𝒓𝒆𝒂

Green’s Theorem

General Form

Vector Form

• =𝑪𝒊𝒓𝒄𝒖𝒍𝒂𝒕𝒊𝒐𝒏

𝑼𝒏𝒊𝒕 𝑨𝒓𝒆𝒂

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Parametrizing Surfaces

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Common Surfaces

Surface Integrals

±

Vector (Flux) Surface Integral

Scalar Surface Integral

Scalar Line Integral

Vector Line Integral

Vector Line Integral (Flux)

Line Integrals

Conservative Vector Fields

A vector field F on domain D is conservative if:

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0• and D is simply connected

Path independence:

𝐶 𝐹 ∙ 𝑛 ⅆ𝑆 = 𝑎

𝑏𝑣(𝑟 𝑡 ) ∙ 𝑁 𝑡 ⅆ𝑡, 𝑁 𝑡 =< −𝑦′ 𝑡 , 𝑥′ 𝑡 >

𝐴𝑟𝑒𝑎 𝑃𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚 =

𝑉𝑜𝑙𝑢𝑚𝑒 𝑃𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑝𝑖𝑝𝑒ⅆ =

𝐹𝑙𝑢𝑥 = 𝑆𝒏 ∙ 𝒗

Applied Vector Geometry

Divergence Theorem

• =𝑭𝒍𝒖𝒙

𝑼𝒏𝒊𝒕 𝑽𝒐𝒍𝒖𝒎𝒆