Final Exam Formula Sheet
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Transcript of Final Exam Formula Sheet
Quadratic Surfaces
Multiple Integration
Average Value
Riemann Sum
Jacobiansβ’ Polar:
β’ Cylindrical:
β’ Spherical:
Mean Value Theorem
Center of Mass
π¦ = ππ₯
π¦ = ππ₯2
π¦ = π π₯
β’ Rewrite as product of limits:
β’ Lines to Test
β’ Convert to Polar if
β’ Plug in values
β’ Squeeze Theorem
Limits
Limit Definitions
Equation of a Tangent Plane
Linear Approximation
Chain Rule for Paths
Chain Rule (Generalized)
Directional Derivatives
Multivariable Differentiation
Optimization
β’
β’
Global Optimization
Second Derivative Test
Lagrange Multipliers
Vector-Valued Functions
Properties
Arc Length
Tangent Line Parametrization
Flux
Stokesβ Theorem
Surface Independence
β’ πͺπππ =πͺππππππππππ
πΌπππ π¨πππ
Greenβs Theorem
General Form
Vector Form
β’ =πͺππππππππππ
πΌπππ π¨πππ
β’
Parametrizing Surfaces
β’β’
β’
β’
β’β’
β’
β’
β’
β’
Common Surfaces
Surface Integrals
Β±
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:
β’
0β’ and D is simply connected
Path independence:
πΆ πΉ β π β π = π
ππ£(π π‘ ) β π π‘ β π‘, π π‘ =< βπ¦β² π‘ , π₯β² π‘ >
π΄πππ πππππππππππππ =
ππππ’ππ πππππππππππππβ =
πΉππ’π₯ = ππ β π
Applied Vector Geometry
Divergence Theorem
β’ =ππππ
πΌπππ π½πππππ