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Calculus III
Washington University in St. Louis
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L24 Math 233 Lecture Notes - Lecture 17: Unit Vector, Directional Derivative, Partial Derivative

L24 math 233 - lecture 17 - chain rule and dependency. Explanation of the chain rule for a function f of two variables composed of function r with one

L24 Math 233 Lecture Notes - Lecture 18: Level Set, Directional Derivative, Implicit Function

L24 math 233 lecture 18- directional derivatives and gradient vectors. If f(x,y) is differentiable, then f(x,y) = 0 defines y = y(x) implicitly. This m

L24 Math 233 Lecture Notes - Lecture 19: Level Set, Nissan L Engine, Unit Vector

L24 math 233 lecture 19- applications of the gradient vector. Recall: then f(x,y) is continuous at every point in the plane. 0 + h y0 + h f. )]/h h 0:

L24 Math 233 Lecture Notes - Lecture 20: Maxima And Minima, Nissan L Engine, Determinant

L24 math 233 lecture 20- maxima and minima. Equations for different types of 3d shapes for level curves of equations with 3 variables. Ax+by+cz+d is a

L24 Math 233 Lecture Notes - Lecture 21: Saddle Point, Fxx, Nissan L Engine

L24 math 233 lecture 21- maxima and minima continued. Maximum and minimum has a tangent line of 0 or no tangent line. Critical points aren"t always max

L24 Math 233 Lecture Notes - Lecture 23: Lagrange Multiplier, Nissan L Engine, Linear Independence

Find critical points i. e. find where the gradient vector = 0. Check the values of f and the second derivatives f f xx yy f xy. For maximum and minimum

L24 Math 233 Lecture Notes - Lecture 24: Lagrange Multiplier, Level Set, Nissan L Engine

L24 math 233 lecture 24- lagrange multipliers continued. Find the maximum/minimum of f(x,y) subject to the constraint g(x,y) = f by these steps: find x

L24 Math 233 Lecture Notes - Lecture 25: Riemann Sum, Nissan L Engine, Iterated Integral

L24 math 233 lecture 25- integration for functions of two variables. Integral of f(x) = area under the graph. This area can be approximated with rectan

L24 Math 233 Lecture 26: L24 Math 233 Lecture 26- Iterated Integrals

Double integrals: d b a c (x, ) y is the function and. [a,b]x[c,d] is the rectangle r in which f is defined is the area of element dxdy. Method 1 (x fi

L24 Math 233 Lecture Notes - Lecture 27: Glossary Of Arithmetic And Diophantine Geometry, Iterated Integral, Nissan L Engine

L24 math 233 lecture 27- iterated integrals and different types domains. This is an iterated integral and represents the volume under the graph (x, )da

L24 Math 233 Lecture Notes - Lecture 28: Cartesian Coordinate System, Multiple Integral, Iterated Integral

To convert between rectangular and polar coordinates, use these equations not (x,y) r, cos sin x = r y = r. R2 = x2 + y2 an t y. D = { r 1 2 r2 d . D =

L24 Math 233 Lecture Notes - Lecture 29: Iterated Integral, Polar Coordinate System, Nissan L Engine

( 1 2 + y2 = 1. L24 math 233 lecture 29- applications of polar coordinates. Find the volume of the solid bounded by the xy plane, the surface inside th

L24 Math 233 Lecture Notes - Lecture 30: Volume Element, Uv Mapping, Parallelogram

L24 Math 233 Lecture Notes - Lecture 31: Parallelogram, Nissan L Engine

Google docs doesn"t allow me to type matrices, so anytime there is a metrix, the. Note: top row will be distinguished from the bottom row with a divisi

L24 Math 233 Lecture Notes - Lecture 32: Nissan L Engine, Linear Map, Cross Product

L24 math 233 lecture 32- jacobians continued and surface area integral. V v v y f (x(u, ), (u, ))(jacobian)dudv. U: write the jacobian as , explore som

L24 Math 233 Lecture Notes - Lecture 33: Iterated Integral, Nissan L Engine, Jacobian Matrix And Determinant

, )dzdydx y z b a (x, q p d c dv f f. D = { x 1 0 y 2 0 z y. Now, we"ll consider how to find formula for d given some other description. Call this a ty

L24 Math 233 Lecture Notes - Lecture 34: Cylindrical Coordinate System, Iterated Integral, Volume Element

L24 math 233 lecture 34- cylindrical and spherical coordinates. Inversion formulas: cos sin x = r y = r. Z = z r = x2 + y2 y. Examples of graphs in cyl

L24 Math 233 Lecture Notes - Lecture 35: Nissan L Engine

(x,y) = <p(x,y),q(x,y)> where p and q are (x, y), A vector field in 2-d (the plane) is a function f functions of two variables. (x,y) = y > = < p (x, )

L24 Math 233 Lecture Notes - Lecture 36: Nissan L Engine

L24 math 233 lecture 36- vector fields continued and line integrals. Not every vector field is a gradient field. Necessary condition: if then derivativ

L24 Math 233 Lecture Notes - Lecture 40: Nissan L Engine, Antiderivative, C More Entertainment

Set up: let c be a smooth curve with a differentiable, 1-1 parametrization r(t), so. Let f(x,y) be a continuous function on an open domain d that conta

L24 Math 233 Lecture Notes - Lecture 41: Nissan L Engine, Simply Connected Space

Is a vector field on an open, connected region d and f has for some potential function on. Theorem: if some path independent line integrals, then f = f

L24 Math 233 Lecture Notes - Lecture 42: Simply Connected Space, Curve, Nissan L Engine

= i is a vector field on d with for some on d) L24 math 233 lecture 39- green"s theorem and vector fields. Theorem: if d is an open, connected, and sim

L24 Math 233 Lecture Notes - Lecture 43: Jordan Curve Theorem, Disjoint Sets, 2D Computer Graphics

L24 math 233 lecture 40- jordan theorem and introduction to green"s theorem. The domain in 2d space is: open if every (x,y) in d is the center of a dis

L24 Math 233 Lecture Notes - Lecture 44: Piecewise, Simply Connected Space, Curve

L24 math 233 lecture 44- proof and application of green"s theorem. Theorem: if c is a simple, piecewise, smooth, closed curve oriented positively, and

L24 Math 233 Lecture Notes - Lecture 45: Nissan L Engine, Multiple Integral, Ellipse

L24 math 233 lecture 45- green"s theorem applications. 1 (x, ) y y = 2 and. Q y = 2 (x, ) x then. Application : find area of an ellipse with major axis