Math 147 - Calc III
Review for Test 1
Test 1 will consist in 5 problems. You are allowed one page of formulas, but with no solutions to homework problems. The page should be turned in, together with the exam.
Please review the theory first, and work the suggested problems. Then continue with the other homework problems, and review the proofs of the theorems.
Ch. 12
Coordinate systems and terminology
- Distance formula and equation of the sphere;
Vectors
- definition, components, magnitude and direction;
- operations with vectors (+, scalar multiplication), properties;
- unit vectors, standard basis;
- resultant force (vector addition).
The dot product
- definition: “polar” and Cartesian forms; properties (bilinear, symmetric; lengh);
- scalar and vector projections; decomposition of a vector relative a given vector;
The cross product
- definition: “polar” and Cartesian forms; properties (bilinear, NOT assoc., NOT comm.);
- triple products and volume.
Equations of lines and planes
- vector, parametric and symmetric equations of a line; direction numbers;
- scalar and vector equations of a plane; normal vector;
- distance formulas (point-to-line, point-to-plane);
Cylinders and quadratic surfaces
- Cylinders
- Quadric surfaces (identify via traces; graph with TI-89; identify equations).
Math 147 - Calc III
Review for Test 2
Test 2 will consist in 5 problems. You are allowed one page with formulas, but with no homework solutions. The page should be turned in, together with the exam.
Please review the theory first, and work the suggested problems. Then continue with the other homework problems, and review the proofs of the theorems.
Ch.13 Vector Functions
Vector-valued functions: Values are vectors depending on 1 parameter (space curve), or more;
Derivatives and integrals of VVF: linear/component-wise operations;product rules (dot & cross-product)
Arc length and curvature: , k =dT/ds, T unit tangent.
Motion in space: velocity and acceleration
- velocity and acceleration;
- decomposing acceleration along velocity: tangential and normal acceleration;
- speed v, radius of curvature R and curvature k=1/R.
- binormal and the “moving frame” (C.S.).
Ch.14 Partial Derivatives
Functions of several variables: definition, level curves/surfaces, contour map.
Limits and continuity: definitions, interpretation, rules and methods to compute limits.
Partial derivatives:
- definition, notations, rule for computing, interpretation (2..n variables);
- higher derivatives, Clairaut’s Th. (fxy=fyx), solutions of PDE.
Tangent planes and linear approximations :
- tangent planes: scalar and vector equation; gradient / slope;
- differentials: scalar and vector form;
- tangent plane to parametric surfaces.
The chain rule:
- Tree diagrams and the corresponding chain rule
- Implicit differentiation, Implicit Function Theorem
Directional derivative and the gradient vector
- Directional derivative: definition as a rate of change in the direction of u, interpretation as a slope of a vertical trace in the direction of u; relation with gradient and differential
- Gradient: definition, relation with directional derivative (“proportionality vector” rel. scalar product), relation with the direction of the maximum slope; orthogonal to level curves/surfaces:
- Tangent planes to level surfaces, normal ~ gradient
Maximum and Minimum:
- local/relative extrema, global/absolute extrema, critical points
- Finding abs. max/min on a closed domain
Lagrange multipliers: one and two constraint equations.
Math 147 - Calc III
Review for Test 3
Test 3 will consist in 5 problems. You are allowed one page with formulas, but with no explicit solutions. The page should be turned in, together with the exam.
Please review the theory first, and work the suggested problems. Then continue with the other homework problems, and review the proofs of the theorems.
Ch.15
1) Double integral
- Definition (rectangles and general regions), properties, Midpoint rule, average value
- Partial integration, iterated integrals, Fubini’s theorem for double integrals:
- Simple regions (V/H, type 1 / 2 or x/y)
- Average value
- Polar rectangle, integration in polar coordinates
2) Triple integrals:
- Definition, simple regions: box, type n (x,y,z), Fubini’s theorem 3=2+1
- Cylindrical coordinates: solids (type x,y,z with D polar rectangle or r-simple);
Transformation from polar to cartesian;
- Spherical coordinates: solids (wedge, or r-simple), transformation from spherical to Cartesian, limits of integration in spherical coordinates, volume element in spherical coordinates.
3) Applications (double or triple integration): 12.5:3, 13, 17;
- Mass and density, moments about coordinate planes, inertia about axes, probabilities
- Surface area: parametric surfaces, graphs of f(x,y), surfaces of revolution.
4) Change of variables in multiple integrals: Jacobian, change of variables formula.
Math 147 - Calc III : Review for Final Test
Final Test will consist in 5 problems (like for Tests 1,2 and 3), together with 5 easy problems or questions. In addition to the pages for Tests 1,2,3, you are allowed one page with formulas, but with no explicit solutions. The pages with formulas should be turned in, together with the exam.
Please review the theory first, and work the suggested problems. Continue working the other homework problems, and review the proofs of the theorems.
For chapters 12-15 see the previous Review sheets for Tests 1,2,3.
Ch.16 Vector Calculus
Ch.16
16.1: Vector fields
- Vector fields (compare with direction fields);
- Examples: velocity, gravitational field, electric force field;
- Gradient fields (recall gradient vector), potential function and conservative vector fields.
16.2: Line integrals
- Curves in dimension 2 (plane) & 3(space);
- Functions (“scalar fields”) and vector fields
- Applications: mass, work done by a force, moments of inertia
16.3: The FTC for Line integrals
- Theorem 2;
- Relation with the differential and the boundary of the curve: .
- Independence of path; characterizing conservative fields;
- Connected, simply connected regions;
- Conservation of energy.
16.4: Green’s Theorems
- Tangential form: circulation – curl
- Normal form: flux-divergence
16.5 Surface Area and Surface Integrals
- Level surface f(x,y,z)=0 projected on the xy-plane: dS=1/(n.k) dxdy
16.6 Parameterized Surfaces
- Parameterization r(u,v):D->R^3; examples: cone, sphere;
- Surface area element: dS=|dr/du x dr/dv| du dv.
16.7 Stokes Th. (Green’s Th. – normal form, in 3D): circulation = flux, curl(F)=Del x F
16.8 Divergence Theorem: Flux(Surface)=Divergence (Volume), div(F)=Del . F; Gauss law.