Section 9.4 Areas and Lengths in Polar Coordinates
2010 Kiryl Tsishchanka
Areas and Lengths in Polar Coordinates
In this section we develop the formula for the area of a region whose boundary is given by a polar equation.
We need to use the formula for th
HW #6
Practice. These exercises are recommended for the further development of your skills.
12.1: 9,11,2530
12.2: 2143 (odd)
Written. Please give complete, well-written solutions to the following exercises.
(1) Show that if R = [a, b] [c, d] and f (x, y
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Read proof at home RREF doc
Is a unique solution only with rref and is infinitely many solutions with
ref?
Substitute parameters for the non leading columns to then
compose a system of equations that is general (meaning, that if I
put a value for the par
with Open Texts
A First Course in
LINEAR ALGEBRA
Lecture Notes
by Karen Seyffarth
Matrices: Matrix Arithmetic
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Matrices: Matrix Arithmetic
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Chapter
When we reduce a system of linear equation to REF or RREF and we
have a row of 0s then that means we have a free parameter that can
take on any value! This means that we have infinite solutions to the
system, as any value put in place of that parameter ca
Matrix with size m n m = rows n = columns
A zero matrix all entries are 0
Row matrix is just one of the rows of the matrix
A column matrix is similar
A row/column matrix is simply called a vector
A square matrix is an even number of rows and columns
The
Our goal is to get as many 0s as possible to make the system as simple
as possible to solve.
Every ele row operation can be reversed.
We want the column of 0s on the lef
The leading ones have to be to the right of the previous
If you have anything abov
MATH 1021 B
Test #1
October 11, 2016
Last Name:_
First Name:_
Student Number:_
Instructions:
1.
2.
3.
4.
Duration 70 minutes.
No calculators permitted.
Show ALL your work.
Use pen to fill-in the cover page. If you use pencil for your solutions you may not
Elementary matrices
A matrix obtained from an identity matrix by performing a single
elementary row operation.
So an operation performed on a matrix is a matrix itself. Each operation
modifies the original and is the new matrix
HW #5
Practice. These exercises are recommended for the further development of your skills.
11.4: 5,15,17,27,29,31
11.5: 1727 (odd)
Written. Please give complete, well-written solutions to the following exercises.
(1) The pressure, volume, and temperatu
HW #10
Practice. These exercises are recommended for the further development of your skills.
13.6: 1122
13.7: 513 (odd)
Written. Please give complete, well-written solutions to the following exercises.
(1) Find the surface area of each surface below.
(a
HW #8
Practice. These exercises are recommended for the further development of your skills.
12.3: 1317 (odd), 20, 29
12.5: 3440, 46 (even)
12.6: 1523 (odd)
Written. Please give complete, well-written solutions to the following exercises.
(1) Use Fubini
MAC 2313
Section 14.1
Keywords:
Domain, level curves, and level surfaces, elevation curves, steepest ascent and descent, contour plots
Domain of a Function:
A function of two variables z fx, y feeds on inputs and produces ouputs. The independent variables
MAC 2313
Section 14.4 Chain Rule and Curves on Surfaces.
Keywords:
Chain Rule, Curves on Surfaces, Function Composition, Curves on Surfaces, Slope on a surface while
following a curve.
The Chain Rule in Calculus I:
Functions are combined with other functi
MAC 2313 Section 14.5
Directional Derivatives and Gradient Vectors
Keywords:
Directional Derivatives, Gradient Vector, Steepest Ascent and Descent, Del Operator.
The Gradient Vector
First, let us introduce , the "Del" differential operator. By itself, thi
MAC 2313 Section 15.3 Double Integration in Polar Coordinates
Keywords:
Rotational Symmetry, Jacobian, Trnasformation Coordinates Formula, Polar Form.
Introduction:
As far as we are concerned the main point of this section is to switch to polar coordinate
MAC 2313 Section 15.1 Double Integration
Keywords:
Double Integration, Integrand, "Roof" Function, "Footprint of Building", Region of Integration, Limits of
Integration, Order of Integration, Red Line, Region of Integration with Curved Boundaries, Solving
MAC 2313
Section 14.7
Extreme Values and Saddle Points of 2D Surfaces
We are not going to mention everything on the subject but look at part of it. You may recall, from Calculus I
that local extrema are associated with critical values and "domain of study
MAC 2313 Coordinate Relations, Mappings, Parametric Representation of Surfaces, and Triple
Integration in Cylindrical and Spherical Coordinates
Keywords: Cartesian, Cylindrical, Spherical Coordinates.
A) Coordinate Systems
Considering the picture above le
MAC 2313 Section 16.1 Line Integrals
Keywords: Line Integral, Length of Curve, Parametric Curve Representation, Parameterization, Linear
Density.
Length of a 3D Curve (Repeat): Extremely intuitive (as opposed to rigorous) explanation of the process:
We ar
MAC 2313
Section 14.3
Keywords:
Slopes on Surfaces, Steepest Ascent, Steepest Descent, Tangent Plane, First Order and Second Order
Partial Differentiation, Directional Derivative, Mixed (Crossed) Derivatives, Continuity, Differentiability,
Corner Point.
B
MAC 2313
Section 14.6: Tangent Plane to 3D Surface
Keywords:
Explicit vs. Implicit Functions, Point of Tangency between 3D Surface and Plane, Tangent Plane, Gradient
Vector, Level Surface.
Introduction:
In this section, we will limit ourselves to the dete
MAC 2313
Section 15.4 Triple Integrals in Rectangular Coordinates
A triple integral which could look as follows,
xb ygx znx,y
xa yfx
zmx,y fx, y, zdzdydx, is not so daunting if you
recall that, once you have managed the inner integral, the rest is a not
HW #1
Practice. These exercises are recommended for the further development of your skills.
10.1: 11, 13, 15
10.6 (p. 558): 21, 23, 29, 31
10.2: 919 (odd), 27
10.3 (p. 536): 1725 (odd)
Written. Please give complete, well-written solutions to the follo
Name:
Calculus III Practice Problems for Final
The Final will cover all material from the course. Especially important topics: gradient/directionalderivative/divergence/curl, maxima/minima problems (including Lagrange multipliers), double and triple
integ