Review Sheet 3 Math H263

[n] means the topic is found on acetate n.

Italics indicate material that could be on the final exam but that was not on the midterms.

Final exam goes through acetate [215].

Tools

Normal vectors to planes in three-dim space

How to tell if planes are parallel or at right angles by considering their normal vectors

Properties of line integrals [127-128]

Differentiation of integrals [142-143]

Green's Theorem [140]

Dot products and cross products of vectors [throughout]

Determinants [107-109]

Definitions (know them and be able to write them correctly)

Level surfaces of functions [3, 6]

Domain of a function [7-9]

Continuous function [10-11]

Partial derivatives [15-16]

Directional derivatives [20]

Unit vector [32]

Gradient [32-33.0]

Tangent plane to a surface at a point on the surface [37]

Critical point [51]

Double integrals [67]

Region [61]

            Horizontally simple [73.0-73.1, 76]

            Vertically simple [76]

            Simple [76]

Triple integrals [98]

Cylindrical coordinates [102, 108]

Spherical coordinates [104-106, 109]

Line integral [119]

Scalar field [126]

Vector field [126]

Conservative field and its characterizations [125,129-130, 137-140]

Jacobian of a mapping [159-160]

Linear mapping [164]

Surface integral [168-170]

Flux density [188]

Divergence of a vector field [191]

Geometry

            Related to partial derivatives [15, 19.0-19.3]

            Related to directional derivatives [20-22, 30-31]

            Related to gradient [38]

            Tangent vector to a curve in space [34.2-37]

            Tangent plane to a surface at a point on the surface [33.1, 37]

Tangent vector in the tangent plane for a curve on a surface [34.2-37]

Normal vector to a tangent plane of a surface [39-40]

Gradient at a point is perpendicular to the level surface through the point [39-40]

Related to LaGrange multipliers [57.0-57.2]

Related to double integrals [63-66]

Double integral in polar coordinates [85-89]

Be able to sketch figures in 2 and 3 dimensions [throughout]

Triple integrals in cylindrical and spherical coordinates [102-109]

 

Examples

 

         Functions of several variables [2]

 

            Partial derivatives [16-18]

 

            Directional derivatives [33.0-33.1, 37.1-37.2]

 

            Chain rule [44-46, 48-50]

 

            Implicit differentiation [49]

 

            Max-min in several variables [54-57.0, 58-60.1]\

 

            Regions [62, 73.0-73.1, 76]

 

            Double integral [69-72]

 

            Iterated integrals [74-75, 77, 79-83, 90-92]

 

            Triple integrals [99-101, 103, 105-109]

 

Work as motivation for line integrals [118-119, 124-125]

 

         Line integral [122-123] and [200.0-200.1]

 

         Vector field [136.1]

        

Simple closed curves [141.1 on 0506]

 

Green’s theorem [148-149] and [201]

 

Green’s theorem for general regions [146, 147.1]

 

Change of variables and Jacobians [151, 153, 161-165, 180]

 

Surface integrals [172-174]

 

Conservative vector fields [211, 215]

 

Surface integral [172-174]

 

Closed orientable surfaces [193]

 

Divergence theorem [198-199, 204]

 

Stoke's Theorem [212-214]

 

 

Facts

 

z = f(x, y) defines a surface in three-dimensional space [throughout]

 

            f(x, y) = c (constant) defines a curve in the plane [throughout]

 

f(x, y, z) = c (constant) defines a surface in three-dim space [throughout]

 

Usually the order in which mixed partial derivatives are calculated makes no difference [18]

 

The tangent plane to the level surface of a function is perpendicular to the gradient. [39-40]

 

Chain rule for several variables [43, 47]

 

Second derivative for max-min in two variables [53]

 

About double integrals [78-79]

 

cos2(theta) = (1/2)(1 + cos(2*theta)

 

sin2 (theta) = (1/2)(1 - cos(2*theta)

 

Line integrals [136.0, 137]

 

 

 

Theorems--with proofs

 

f(x, y) = x + y is continuous [12]

 

f(x, y) = xy is continuous [13-14]

 

Theorems--no proofs (know what they mean and be able to state and use them correctly)

Big delicate theorem

            Analogous result for functions of one variable [23-24]

            Motivation and statement of the theorem [22-24]

            Statement of the theorem [25]

            Proof [26-28]

            Generalization to three variables [29]

 

The directional derivative of f(x, y) in the direction of the unit vector u = ai + bj is afx(x, y) + bfy(x, y) [31]

Proofs of the characterizations of conservative fields [131-135, 140]

Green’s theorem [140, 143-146]

Gauss' Theorem(Divergence Theorem) [192, 194-197]

Stoke's Theorem [205-210] 

Calculations

 

            Distance formula in n-dimensional space [throughout]

 

            Use of the dot product to find angles in n-dim space [throughout]

 

            Partial derivatives [16-18]

Normal vector to a tangent plane of a surface [39-40]

Equation for a tangent plane of a surface [40-42]

about tangent planes of a surface [throughout]

gradient [32-33.0]

directional derivatives [31-33.0, 37.1-37.2]

chain rule for several variables [34.0, 43, 47]

implicit partial differentiation [49-50]

max-min in two variables

            second derivative test [53]

substitution [55-56]

            Lagrange multipliers [57.0-57.2]

Iterated integrals [74-75, 79-83]

Applications of double and triple integrals [82-84, 100-101] and [202]

Volumes as double integrals [93-97]

Double integrals in polar coordinates [85]

Triple integrals [99-101]

Triple integrals in cylindrical coordinates [102-103, 108]

Triple integrals in spherical coordinates [104-106, 109]

Surface area

         As a double integral [110-114, 116-117.4]

         For surfaces given parametrically [183-186]

Line integrals [120-121]

Using Green's Theorem [148-149]

Change of variables (substitution) in integrals in dimensions greater than one. [157-163]

Change of variables and Jacobians [150-156, 175-176, 178]

Parametric representations of surfaces [166-167]

Determinants [107-109, 176-177]

Surface integrals [171, 187]

Fluid flow across a surface [188-190, 192] 

Notation

         for partial derivatives [16]

            for gradient [32]

            for multiple integrals [68]

            for iterated integrals [72]

            for triple integrals [98]

         for line integrals [119, 121]

            for Jacobians [107-109]

         for surface integrals [171]

         use of the 'del'-operator [203, 209]