pasmith's latest activity
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Ppasmith replied to the thread High School Uncertain about part of the Order of Operations, from a formula.Exponentiation has precedence over multiplication.
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Ppasmith reacted to wrobel's post in the thread Solving an elementary trigonometric equation with
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Perhaps it would be clearer to write: $$\sin x=a,\quad |a|\le 1\Longrightarrow$$ $$x=\arcsin a+2\pi n$$ and $$x=\pi-\arcsin a+2\pi... -
Ppasmith replied to the thread Undergrad Is the Cauchy stress tensor really a tensor?.If ##T : V^2 \to \mathbb{R}## is a rank 2 tensor following Jeevanjee's definition, then for any ##w \in V## we can define a linear map...
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Ppasmith replied to the thread Solving an elementary trigonometric equation.$$n\pi + (-1)^n \frac{\pi}4 - \frac{\pi}3 = n\pi + \frac{\pi}{12} (3(-1)^n - 4).$$ For even ##n = 2m## this is ##2m\pi -...
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Ppasmith replied to the thread Undergrad Best way to numerically solve this system of equations?.It may be easier to express the first two as $$ (1 - sN_i)(1 - c_i \lambda) + sN_i = 1 - c_i\lambda + sc_iN_i \lambda = 0.$$ Newton's...
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Ppasmith replied to the thread Undergrad Why solve J(X_k) ΔX_k = -F(X_k) instead of computing J⁻¹(X_k) directly.That appears to be the only method Excel provides for solving ##Ax = b##. The documentation does not explain what method Excel uses to...
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Ppasmith replied to the thread Great circle between two points on the Earth's surface.I would rotate the sphere so that one point is at the north pole (0,0,1). Then ##\theta## is just the colatitude of the image of the...
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Ppasmith replied to the thread Undergrad Why solve J(X_k) ΔX_k = -F(X_k) instead of computing J⁻¹(X_k) directly.If ##A## is invertible, then solving ##Ax = b## by Gaussian elimination where ##b## consists of zeros except for a 1 in the ##i##th row...
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Ppasmith replied to the thread Undergrad Why solve J(X_k) ΔX_k = -F(X_k) instead of computing J⁻¹(X_k) directly.We don't solve linear systems numerically by finding an inverse matrix; that is incredibly inefficient. Instead we use other methods...
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Ppasmith replied to the thread Finding the range of k for three real roots of a cubic.Your working appears to be correct, and I obtained the same result by considering the discriminant of ##f##. What did you do to verify...
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Ppasmith replied to the thread Undergrad Do non-orthogonal coordinate systems mean dependent coordinates?.Coordinates are independent, whether orthogonal or nonorthogonal. Introducing such a dependency restricts you to a curve or a surface...
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Ppasmith replied to the thread Undergrad Limit behaviour of Fibonacci sequence.Alternatively, we can also consider that the general solution of ##F_{n+2} = F_{n+1} + F_n## is $$\begin{split} F_n &= A \left( \frac{1...
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Ppasmith replied to the thread Undergrad Limit behaviour of Fibonacci sequence.A slight correction is indeed required. From ##a_n \geq 1## we have ##a_n \varphi \geq \varphi ## so that $$ |a_{n+1} - \varphi| \leq...
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Ppasmith replied to the thread Undergrad Limit behaviour of Fibonacci sequence.There is a much easier way to arrive at these results. From ##a_{n+1} = 1 + a_n^{-1}## and the definition of ##\varphi## we can obtain...
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Ppasmith replied to the thread Undergrad Why should we need to re-prove theorems that have been proved already?.So that we understand the proofs, and can satisfy ourselves that they are indeed correct proofs. We don't take other people's word for...