Linear Approximations and Differentials - Revision history

Khanh at 05:45, 14 November 2021

2021-11-14T05:45:37Z

Lila: /* Resources */

2021-10-28T15:46:56Z

Resources

Lila at 19:47, 18 October 2021

2021-10-18T19:47:19Z

Lila: /* Resources */

2021-10-18T19:45:41Z

Resources

Lila at 19:44, 18 October 2021

2021-10-18T19:44:58Z

Khanh at 21:25, 6 October 2021

2021-10-06T21:25:58Z

Khanh at 21:25, 6 October 2021

2021-10-06T21:25:11Z

James.kercheville: Added video links

2020-08-21T22:18:18Z

Added video links

James.kercheville: Added links for PowerPoint and worksheets

2020-08-14T20:34:30Z

Added links for PowerPoint and worksheets

New page

* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Linear%20Approximation%20and%20Differentials/MAT1214-4.2LinearizationAndDifferentialsPwPt.pptx Linearization and Differentials] PowerPoint file created by Dr. Sara Shirinkam, UTSA.

* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Linear%20Approximation%20and%20Differentials/MAT1214-4.2LinearizationAndDifferentialsWS1.pdf Linearization and Differentials Worksheet 1]

* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Linear%20Approximation%20and%20Differentials/MAT1214-4.2LinearizationAndDifferentialsWS2.pdf Linearization and Differentials Worksheet 2]

← Older revision		Revision as of 05:45, 14 November 2021
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	* [https://en.wikipedia.org/wiki/Differential_(infinitesimal) Differential (infinitesimal), Wikipedia] under a CC BY-SA license		* [https://en.wikipedia.org/wiki/Differential_(infinitesimal) Differential (infinitesimal), Wikipedia] under a CC BY-SA license
	* [https://en.wikipedia.org/wiki/Linear_approximation Linear approximation, Wikipedia] under a CC BY-SA license		* [https://en.wikipedia.org/wiki/Linear_approximation Linear approximation, Wikipedia] under a CC BY-SA license
−
−	~~==References==~~
−	~~# "12.1 Estimating a Function Value Using the Linear Approximation". Retrieved 3 June 2012.~~
−	~~# Lipson, A.; Lipson, S. G.; Lipson, H. (2010). Optical Physics (4th ed.). Cambridge, UK: Cambridge University Press. p. 51. ISBN 978-0-521-49345-1.~~
−	~~# Milham, Willis I. (1945). Time and Timekeepers. MacMillan. pp. 188–194. OCLC 1744137.~~
−	# Nelson, Robert; M. G. Olsson (February 1987). "The pendulum – Rich physics from a simple system" (PDF). American Journal of Physics. 54 (2): 112–121. Bibcode:1986AmJPh..54..112N. doi:10.1119/1.14703. Retrieved 2008-10-29.
−	~~# "Clock". Encyclopædia Britannica, 11th Ed. 6. The Encyclopædia Britannica Publishing Co. 1910. p. 538. Retrieved 2009-03-04. includes a derivation~~
−	~~# Halliday, David; Robert Resnick; Jearl Walker (1997). Fundamentals of Physics, 5th Ed. New York: John Wiley & Sons. p. 381. ISBN 0-471-14854-7.~~
−	~~# Cooper, Herbert J. (2007). Scientific Instruments. New York: Hutchinson's. p. 162. ISBN 1-4067-6879-0.~~
−	~~# Ward, M. R. (1971). Electrical Engineering Science. McGraw-Hill. pp. 36–40. ISBN 0-07-094255-2.~~

@@ Line 80: / Line 80: @@
 * [https://youtu.be/FYKY-3JSzfw Cal1 Differentials Part2] video lecture by Instructor Beatty, UTSA.
 * [https://youtu.be/3hoiQJhzH3Y Cal1 Differentials Part3] video lecture by Instructor Beatty, UTSA.
 ==References==

@@ Line 28: / Line 28: @@
 ===Applications===
-====Optics====
+'''Optics'''
-====Period of oscillation====
+Gaussian optics is a technique in geometrical optics that describes the behavior of light rays in optical systems by using the paraxial approximation, in which only rays which make small angles with the optical axis of the system are considered. In this approximation, trigonometric functions can be expressed as linear functions of the angles. Gaussian optics applies to systems in which all the optical surfaces are either flat or are portions of a sphere. In this case, simple explicit formulae can be given for parameters of an imaging system such as focal distance, magnification and brightness, in terms of the geometrical shapes and material properties of the constituent elements.
 The period of swing of a simple gravity pendulum depends on its length, the local strength of gravity, and to a small extent on the maximum angle that the pendulum swings away from vertical, ''θ''<sub>0</sub>, called the amplitude. It is independent of the mass of the bob. The true period ''T'' of a simple pendulum, the time taken for a complete cycle of an ideal simple gravity pendulum, can be written in several different forms, one example being the infinite series:
@@ Line 47: / Line 48: @@
 In the linear approximation, the period of swing is approximately the same for different size swings: that is, ''the period is independent of amplitude''. This property, called isochronism, is the reason pendulums are so useful for timekeeping. Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.
-====Electrical resistivity====
+'''Electrical resistivity'''
 The electrical resistivity of most materials changes with temperature. If the temperature ''T'' does not vary too much, a linear approximation is typically used:

@@ Line 2: / Line 2: @@
 In mathematics, a '''linear approximation''' is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.
-==Definition==
+==Linear Approximation==
 Given a twice continuously differentiable function <math>f</math> of one real variable, Taylor's theorem for the case <math>n = 1 </math> states that
@@ Line 26: / Line 26: @@
 where <math>Df(a)</math> is the Fréchet derivative of <math>f</math> at <math>a</math>.
-==Applications==
+===Applications===
-===Optics===
+====Optics====
 '''Gaussian optics''' is a technique in geometrical optics that describes the behavior of light rays in optical systems by using the paraxial approximation, in which only rays which make small angles with the optical axis of the system are considered. In this approximation, trigonometric functions can be expressed as linear functions of the angles. Gaussian optics applies to systems in which all the optical surfaces are either flat or are portions of a sphere. In this case, simple explicit formulae can be given for parameters of an imaging system such as focal distance, magnification and brightness, in terms of the geometrical shapes and material properties of the constituent elements.
-===Period of oscillation===
+====Period of oscillation====
 The period of swing of a simple gravity pendulum depends on its length, the local strength of gravity, and to a small extent on the maximum angle that the pendulum swings away from vertical, ''θ''<sub>0</sub>, called the amplitude. It is independent of the mass of the bob. The true period ''T'' of a simple pendulum, the time taken for a complete cycle of an ideal simple gravity pendulum, can be written in several different forms, one example being the infinite series:
@@ Line 47: / Line 47: @@
 In the linear approximation, the period of swing is approximately the same for different size swings: that is, ''the period is independent of amplitude''. This property, called isochronism, is the reason pendulums are so useful for timekeeping. Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.
-===Electrical resistivity===
+====Electrical resistivity====
 The electrical resistivity of most materials changes with temperature. If the temperature ''T'' does not vary too much, a linear approximation is typically used:
 :<math>\rho(T) = \rho_0[1+\alpha (T - T_0)]</math>
 where <math>\alpha</math> is called the ''temperature coefficient of resistivity'', <math>T_0</math> is a fixed reference temperature (usually room temperature), and <math>\rho_0</math> is the resistivity at temperature <math>T_0</math>. The parameter <math>\alpha</math> is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, <math>\alpha</math> is different for different reference temperatures. For this reason it is usual to specify the temperature that <math>\alpha</math> was measured at with a suffix, such as <math>\alpha_{15}</math>, and the relationship only holds in a range of temperatures around the reference. When the temperature varies over a large temperature range, the linear approximation is inadequate and a more detailed analysis and understanding should be used.
 ==Resources==

@@ Line 52: / Line 52: @@
 :<math>\rho(T) = \rho_0[1+\alpha (T - T_0)]</math>
 where <math>\alpha</math> is called the ''temperature coefficient of resistivity'', <math>T_0</math> is a fixed reference temperature (usually room temperature), and <math>\rho_0</math> is the resistivity at temperature <math>T_0</math>. The parameter <math>\alpha</math> is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, <math>\alpha</math> is different for different reference temperatures. For this reason it is usual to specify the temperature that <math>\alpha</math> was measured at with a suffix, such as <math>\alpha_{15}</math>, and the relationship only holds in a range of temperatures around the reference. When the temperature varies over a large temperature range, the linear approximation is inadequate and a more detailed analysis and understanding should be used.
 ==References==
@@ Line 62: / Line 71: @@
 # Cooper, Herbert J. (2007). Scientific Instruments. New York: Hutchinson's. p. 162. ISBN 1-4067-6879-0.
 # Ward, M. R. (1971). Electrical Engineering Science. McGraw-Hill. pp. 36–40. ISBN 0-07-094255-2.

@@ Line 71: / Line 71: @@
 ==Resources==
 * [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Linear%20Approximation%20and%20Differentials/MAT1214-4.2LinearizationAndDifferentialsPwPt.pptx  Linearization and Differentials] PowerPoint file created by Dr. Sara Shirinkam, UTSA.
 * [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Linear%20Approximation%20and%20Differentials/MAT1214-4.2LinearizationAndDifferentialsWS1.pdf  Linearization and Differentials Worksheet 1]

@@ Line 1: / Line 1: @@
-* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Linear%20Approximation%20and%20Differentials/MAT1214-4.2LinearizationAndDifferentialsPwPt.pptx  Linearization and Differentials] PowerPoint file created by Dr. Sara Shirinkam, UTSA.
+[[File:TangentGraphic2.svg|thumb|300px|Tangent line at (''a'', ''f''(''a''))]]
-* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Linear%20Approximation%20and%20Differentials/MAT1214-4.2LinearizationAndDifferentialsWS1.pdf  Linearization and Differentials Worksheet 1]
+However, if one takes the linear approximation (i.e. if the amplitude is limited to small swings,) the period is:
-* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Linear%20Approximation%20and%20Differentials/MAT1214-4.2LinearizationAndDifferentialsWS2.pdf  Linearization and Differentials Worksheet 2]
+:<math>T \approx 2\pi \sqrt\frac{L}{g} \qquad \qquad \qquad \theta_0 \ll 1  \qquad (1)\,</math>
-* [https://youtu.be/9fW2QX-ik2I Cal1 Linearization Part1] video lecture by Instructor Beatty, UTSA.
+The electrical resistivity of most materials changes with temperature. If the temperature ''T'' does not vary too much, a linear approximation is typically used:
 * [https://youtu.be/72LAzx9dGXw Cal1 Differentials Part1] video lecture by Instructor Beatty, UTSA.
 * [https://youtu.be/FYKY-3JSzfw Cal1 Differentials Part2] video lecture by Instructor Beatty, UTSA.
 * [https://youtu.be/3hoiQJhzH3Y Cal1 Differentials Part3] video lecture by Instructor Beatty, UTSA.

← Older revision		Revision as of 22:18, 21 August 2020
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	* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Linear%20Approximation%20and%20Differentials/MAT1214-4.2LinearizationAndDifferentialsWS2.pdf Linearization and Differentials Worksheet 2]		* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Linear%20Approximation%20and%20Differentials/MAT1214-4.2LinearizationAndDifferentialsWS2.pdf Linearization and Differentials Worksheet 2]
		+
		+
		+
		+	* [https://youtu.be/9fW2QX-ik2I Cal1 Linearization Part1] video lecture by Instructor Beatty, UTSA.
		+
		+
		+	* [https://youtu.be/72LAzx9dGXw Cal1 Differentials Part1] video lecture by Instructor Beatty, UTSA.
		+
		+	* [https://youtu.be/FYKY-3JSzfw Cal1 Differentials Part2] video lecture by Instructor Beatty, UTSA.
		+
		+	* [https://youtu.be/3hoiQJhzH3Y Cal1 Differentials Part3] video lecture by Instructor Beatty, UTSA.