MATH 324 Advanced Multivariable Calculus
Contents
Double Integrals
Double integrals in Cartesian coordinates
Description | Equations |
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Double integrals | |
Fubini’s Theorem |
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Separation of iterative integrals |
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Type I region |
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Type II region |
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Addition of double integrals | |
Constant multiple of double integrals | |
Region separation of double integrals | |
Area of a region | |
Average value of a function |
Double integrals in polar coordinates
Description | Equations |
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Transformation to polar coordinates | |
Double integrals in polar coordinates |
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Double integrals in general polar region |
Change of variables for double integrals
Description | Equations |
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Transformation of two variables | |
Jacobian of transformation of two variables | |
Change of variables for differentials | |
Change of variables for double integrals |
Applications of double integrals
Description | Equations |
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Density function | |
Mass | |
Moment about x-axis | |
Moment about y-axis | |
Center of mass | |
Moment of inertia about x-axis (second moment) |
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Moment of inertia about y-axis (second moment) |
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Moment of inertia about the origin (polar moment of inertia) |
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Surface area |
Triple Integrals
Triple integrals in Cartesian coordinates
Description | Equations |
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Triple integrals | |
Fubini’s Theorem $B = x\times y\times z = \newline [a, b]\times[c, d]\times[r, s]$ |
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Type 1 region |
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Type 2 region |
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Type 3 region |
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Example of a general region (6 general regions) |
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Volume of a solid |
Triple integrals in cylindrical coordinates
Description | Equations |
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Transformation to cylindrical coordinates | |
Range of cylindrical coordinates | |
Triple integrals in general cylindrical region |
Triple integrals in spherical coordinates
Description | Equations |
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Transformation to spherical coordinates | |
Range of spherical coordinates | |
Triple integrals in general spherical region |
Change of variables for triple integrals
Description | Equations |
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Transformation of three variables | |
Jacobian of transformation of three variables | |
Change of variables for differentials | |
Change of variables for triple integrals |
Applications of triple integrals
Description | Equations |
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Mass | |
Moments about coordinate planes | |
Center of mass | |
Moments of inertia about coordinate axes |
Partial Differentiation
Chain rule
Description | Equations |
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Chain rule |
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Chain rule |
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Chain rule (general) , where |
Directional derivatives and gradient vector
Description | Equations |
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Assumptions | Unit vector Independent variables |
General directional derivatives | |
General gradient vectors | |
General directional derivatives and gradient vectors | |
Directional derivative in 2D | |
Gradient vector and maximum values | where |
Gradient vector tangent vector for level surface |
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Tangent plane in terms of gradient vector (normal vector) | |
Symmetric equation of normal line |
Vector Calculus
Arc length and parameterization of curves
Description | Equations |
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Vector field | |
Conservative vector field and potential function |
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Parameterization of line segments | , for |
Parameterization of circles | |
Parameterization of functions | |
Arc length | |
Arc length parameter |
Line integrals
Description | Equations |
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Line integral | |
Line integral with respect to arc length | |
Line integral with respect to and | |
Line integrals of vector fields | |
Line integrals of vector fields and scalar fields | |
Orientation properties of line integrals with respect to arc length, , and | |
Orientation properties of line integrals of vector fields | |
Line integral of a piecewise-smooth curve |
Fundamental theorem of line integrals
- path - a smooth curve with initial and terminal point
- simple curve - a curve that does not intersect itself anywhere between its endpoints
- closed curve - a curve where its terminal point coincides with its initial point
- simple region - a region that is bounded by two line segments in one direction (type-1, type-2 regions)
- open region - a region that does not contain boundary points
- closed region - a region that contains all boundary points
- connected region - two points in the region can be joined by a path that lies in the region
- simply-connected region - a region that every simple closed curve in D encloses only points that are in D
- has no hole
- doesn’t consist of separate pieces
Description | Equations |
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Fundamental theorem of line integrals | |
Line integrals of non-conservative fields are not path independent (same end points) | |
Line integrals of conservative fields are path independent (same end points) | |
Line integrals of closed path | is path independent |
Path independence and conservative vector field (open, connected region) | is path independent (conservative field) |
Property of conservative vector field | |
Determine conservative vector field in 2D (open simply-connected region) |
Summary
- Fundamental theorem of line integral (FTL) is always true (with assumptions).
- Other statements are not true for general
- They have to be verified for each given or derived from theorems.
Curl and divergence
Description | Equations |
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Gradient | |
Curl | |
Divergence | |
Laplace operator on scalar functions | |
Laplace operator on vector fields | |
Property of conservative vector field | |
Determine conservative vector field in 3D (open simply-connected region) |
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Property of divergence and curl | |
Determine curl field | is not curl of any field |
Green’s theorem
Description | Equations |
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Green’s Theorem (positively oriented, piecewise-smooth, simple, closed curve enclosing ) |
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Circulation-Curl form of Green’s Theorem | |
Flux-Divergence form of Green’s Theorem | |
Area of region enclosed by |
Surface area and parameterization of surfaces
Description | Equations |
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General parametric surface | |
Parametric equation of a plane | |
Parametric equation of a sphere | |
Parametric equation of an explicit function | |
Parametric equation of a surface of revolution | |
Normal vector of a tangent plane | |
Surface area of a parametric surface | |
Surface area of the graph of an explicit function | |
Surface area of the graph of an implicit function |
Surface integral
Description | Equations |
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Surface integral of a function over a parametric surface | |
Surface integral of an explicit function | |
Surface integral of piecewise smooth surface | |
Unit normal vector | |
Surface integral of a vector field over a parametric surface |
Stoke’s theorem
Description | Equations |
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Stoke’s Theorem (S: oriented, piecewise-smooth surface C: simple, closed, piecewise-smooth curve F: continuous partial derivatives in open region) |
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Alternative surface ( is a common curve of and ) |
Divergence theorem (Gauss’s theorem)
Description | Equations |
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Divergence Theorem (E: simple, solid region S: positively oriented surface F: continuous partial derivatives in open region) |
Appendix
Types of functions
Function Type | Domain Range | Equation | Example |
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Function of several variables | |||
Vector-valued function | |||
Vector field |