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# Geometry and Topology

### Metric Tensors and Christoffel Symbols

Problem 1. Derive the formula given below for the Christoffel symbols ?_ij^k of a Levi-Civita connection in terms of partial derivatives of the associated metric tensor g_ij. ?_ij^k = (1/2) g^kl {?_i g_lj ? ?_l g_ij + ?_j g_il }. Problem 2. Compute the Christoffel symbols of the Levi-Civita connection associated to ea

### Find the Area of a Rectangle

For your assignment this week, imagine that you will be building a shed in your back yard. The shed requires a cement foundation that is rectangular in shape. You would like to mark the location of the cement foundation to ensure that it is the correct size and shape. You do not have any special equipment that will help ensure t

### Math for Water Distribution Operator Certificate

See the attached file. I'm taking a Water Supply Technology math class to get a Water Distribution Operator Certificate. We are covering Volume of Rectangular and Cylindrical Tanks, Pipelines, and Rectangular Channels. We have not covered things like flow rate as it relates to time as in detention time. We are not there yet

### Trivial, Discrete, Cofinite, Usual and Sogenfrey Topology

Let X = Y = IR and f: X -> Y be given by f(x) = { x^2 + 1 x >= 0 { 0 x < 0 Consider the following statement A is open in Y then f^-1(A) is open in X. With the following settings on X,Y, determine whether the above statement is true a) X,Y are given the usual topology

### Quantitative Analysis of The Italian General's Pizza Parlour

1. You must show all steps including formulas used and all calculations done to arrive at the final answers. Incomplete solutions will receive partial credit. 2. Use at least four significant digits at all intermediate steps. Round off the final answers appropriately. Note: 0.0042 is only two significant digits as leading zer

### Inventory Management

Shoe Shine is a local retail shoe store located on the north side of Centerville. Annual demand for a popular sandal is 500 pairs, and John Dirk, the owner of Shoe Shine, has been in the habit of ordering 100 pairs at a time. John estimates that the ordering cost is \$10 per order. The cost of the sandal is \$5 per pair. For Jo

### Calculating the volume of concrete in a road

A section of road follows a circular arc with a central angle of 23.8. The radius of the inside of the curve is 281.0 m, and the road is 15.4 m wide. What is the volume of the concrete in the road if it is 0.305 m thick? The volume of the concrete used for the section of road is ___ m^3. Round to five decimal places.

### prove that families are orthogonal

Let the function f(z) = u(x,y) + iv(x,y) be analytic in a domain D, and consider the families of level curves u(x,y) = c1 and v(x,y) = c2. Prove that these families are orthogonal. More precisely, show that if z0 = (x0, y0) is a point in D which is common to u(x,y) = c1 and v(x,y) = c2 and if f'(z0) doesn't equal 0, then the l

### dimensions that maximize the area

If a window entails a seamless glass area formed by a rectangle capped by semi-circle, if the semi-circle's diagonal and the rectangle's width coincide, and if the window's exterior perimeter is 16 feet, determine the dimensions that maximize the window's glass area.

### The Dot Product (computing work)

A force of 80 lbs. on a rope is used to pull a box up a ramp inclined at 10degrees from the horizontal. The rope forms an angle of 33degrees with the horizontal. how much work is done pulling the box 22 feet along the ramp. work done is ? foot-pounds

### Finding the perimeter of a rectangle: Example

Points A, B,C and D are on a square. The area of teh squre is 36square units. Which is true about the perimeter of rectangle ABCD? 1) greater than 24 units 2) less than 24 units 3) equal to 24 units 4) cannot be determined

### QR Factorization: Example Problems

1) Apply the Gram-Schmidt process to a_1 = (0 0 1)^T, a_2 = (0 1 1)^T, and a_3 = (1 1 1)^T and write the result in the form A=QR 2) Apply the Gram-Schmidt process to the vectors above in reverse order: in the form A = QR. where a_1= (1 1 1)^T, a_2=(0 1 1)^T, and a_3=(0 0 1)^T

### QR factorization

Let A= (1 0 -1 and b=(1 1 2 1 1 1 1 -3 1 0 1 1) 1) 1) Determine the QR factorization of A 2) Use the QR factors in 1) to determine the lease squares solution to Ax=b. Ax=b is the solution related to Rx=Q^Tb

### Orthonormal basis

Consider the following three vectors in R^3: x_1=(1, -1, 0, 2) x_2=( 1,1,1,0) x_3= (-1,-1,2,0) a) Verify that {x_1, x_2, x_3} are orthogonal with the standard inner product in R^4 b) Find a nonzero vector x_4 such that {x_1, x_2, x_3, x_4} is a set of mutually orthogonal vectors. c) Convert the resulting set into an ort

### Application of Newton's Laws to Systems of Masses with Ropes

Solve for No. 1 to No. 5 Tension (forces) Tension = ? (See attached file for diagrams)

### The volume of the right circular cone

** Please see the attached file for the diagram ** Find the approximate value of the volume of the right circular cone with a circular base shown below. Approximate your solution to the nearest hundredth. 320 m3 4021.24 m3 4557.40 m3 1005.31 m3 1139.35 m3

### Finding the Volume of a Can

Question: Find the exact value of the volume of a can of asparagus with a diameter of 8 centimeters and height of 10 centimeters. Include correct units with your solution. Use "pi" in place of the pi symbol.

### Finding a Surface Area with Given Length, Width and Height

The following has me really stumped. Can You please help me with this problem?: Find the surface area of the following room measurements: LENGTH:8 feet *10 inches = 106 inches WIDTH: 12 feet * 9 inches = 153 inches HEIGHT: 7 feet * 10 inches = 94 inches Then: A gallon of paint covers about 350 square feet. How many g

### The Volume of the Displaced Water

If a solid object such as small brick is submerged in water, it will displace a quantity of water equal to its volume. This fact can be used as a method for measuring volumes. Suppose that a small brick is put into an aquarium which is the shape of a rectangular solid 1 foot wide, 3 feet long, and 2 feet high. Before the sm

### The change in volume

Water is poured into a funnel at the constant rate of 1 in^3/sec and flows out at a rate of 1/2 in^3/sec. The funnel is a right circular cone with a height of 4 inches and a radius of 2 inches at the base. How fast is the water level changing when the water is 2.5 inches high? Show all work.

### graphing, using substitution, or elimination

1. Systems of equations can be solved by graphing, using substitution, or elimination. What are the pros and cons of each method? 2. What is the difference between intersecting and perpendicular lines? Can two lines exist that are not intersecting or parallel? Explain your answer.

### volume of the solid generated by revolving

Please see attachment. Use the shell method to find the volume of the solid generated by revolving the regions bounded by the curves and lines about the y-axis. y = x^2, y = 7-6x, x = 0, for x >=0

### The Height of a 10 Inch TV

Television Sets: What does it mean to refer to a 20 inch TV set or a 25 inch TV set? Such units refer to the diagonal of the screen. A 10 inch TV set also has a width of 8 inches. What is the height of a 10 inch TV? ______ inches.

### Equation of Circle, Parabola & Ellipse

Find the equations of the following: a) A circle where (4,-1) and (-6,6) are endpoints of a diameter. b) A parabola with a focus at (3,4) and a directrix at x=-1. c) An ellipse with vertices at (-2,5) and (-2,1) and a minor axis 2 units long. Please see the attachment.

### How to find the volume of a pyramid with a rectangular base

Find the volume of the pyramid with a rectangular base. 11 in height, 12 in length, and 10 in width. Choices of answers: a. 120 cubic inches b. 440 cubic inches c. 528 cubic inches d. 336 2/3 cubic inches e. 1320 cubic inches

### Word Problem

It is a common experience to hear the sound of a low flying airplane, and look at the wrong place in the sky to see the plane. Suppose that a plane is traveling directly toward you at a speed of 200 mph and an altitude of 3,000 feet, and you hear the sound at what seems to be an angle of inclination of 20 degrees. At what ang

### Examples of trapezoids

Determine whether it is possible to find a circumscribable, isosceles "true" trapezoid (in which the parallel sides are not congruent) that has each of the following properties. Justify the answer. a) A diagonal bisects an angle of the trapezoid. b) The trapezoid is cyclic c) The diagonals are perpendicular to each other

### Minimization of cost problem

The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials.

### Is R Compact with This Topology?

Please solve the following problem: " Let X be any set with infinite elements ; is it true that A = { A a subset of X : X - A is either finite or countable} U { X,0} is a topology on X ?. Why ? ( note: the 0 means the empty set). Is R compact with this topology?

### Find the Length and Width

The peremiter of a cross section of a piece of lumber is 34 1/2 in. The length is twice the width. Find the actual dimension of the cross section of this piece of lumber. The width is how many inches?? The length is how many inches?? Show work and anwser both questions