Date post: | 10-Apr-2016 |
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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