Show that the following functions are harmonic, that is, that they satisfy Laplace’s equation, and find for each a function f ( z ) of which the given function is the real part. Show that the function v ( x , y ) (which you find) also satisfies Laplace’s equation. xy
Show that the following functions are harmonic, that is, that they satisfy Laplace’s equation, and find for each a function f ( z ) of which the given function is the real part. Show that the function v ( x , y ) (which you find) also satisfies Laplace’s equation. xy
Show that the following functions are harmonic, that is, that they satisfy Laplace’s equation, and find for each a function
f
(
z
)
of which the given function is the real part. Show that the function
v
(
x
,
y
)
(which you find) also satisfies Laplace’s equation.
Fill in the blanks to complete each of the following theoremstatements:
The derivative of a vector function r(t) is given by r'(t) =limh→0__________________.
The Chain Rule for Vector-Valued Functions: Let t = f (τ) bea differentiable real-valued function of τ, and let r(t) bea differentiable vector function with either two or threecomponents such that f (τ ) is in the domain of r for everyvalue of τ on some interval I. Then dr/dτ =_______________ .
Let C be the graph of a twice-differentiable vector functionr(t) defined on an interval I and with unit tangent vector T(t). Then the curvature κ of C at a point on the curve is given by κ = ||__________|| / ||__________|| and κ = ||_____x______||/||_________||3
Let y = f (x) be a twice-differentiable function. Then thecurvature of the graph of f is given by κ =| | / ( )3/2
Let C be the graph of a vector function r(t) = ⟨x(t), y(t)⟩ inthe xy-plane, where x(t) and y(t) are twice-differentiablefunctions of t such that x'(t)…
The position function of a particle is given by ?(?) , where t is measured issecond and s is measured in metres.
a) Find the velocity and acceleration functions.
b) When does the particle change direction?c) What is the velocity at ? = 3 seconds? Is the particle moving away or towards its starting position?
Find the harmonic function w(x, y) = ax2 + bxy + cy2 , in cartesian coordinates, that takes the values cos(2θ) on the unit circle.
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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY