User:Egm6322.s12.team2.steele.m2/Mtg5

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

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Example: Derivative operator

Let $$u (x)$$ and $$v(x)$$ be any 2 functions of x. Let $$\alpha$$ and $$\beta$$ be any 2 real numbers. We write $$\forall u,v: \mathbb R \rightarrow \mathbb R$$ Domain		range

$$x \mapsto u(x)$$ $$x \in \mathbb{R}$$			$$u(x) \in \mathbb{R}$$ Note: Vertical bar	value (image) of u at x

$$x \mapsto v(x)$$ $$\forall \alpha, \beta \in \mathbb{R}$$

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Note: $$\rightarrow$$	set-to-set mapping $$\mapsto$$	point-to-point mapping (end Note)	/// (end Example) /// Example: Matrix algebra Consider the following operator:

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$$\forall \mathbf{u}, \mathbf{v} \in \mathbb{R}^{m \times 1}$$ And any two real numbers, i.e., $$\forall \alpha, \beta \in \mathbb{R}$$ We have the matrix A as a linear operator since:

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Nonlinear = not linear, i.e., not satisfying [[media:Pea1.f11.mtg4.djvu|Eqn(3) p.4-6:]]

Example: In the EOM (1) p. 3-3, the first term $$c_3(Y^1, t)\ddot{Y^1}$$ is nonlinear 2nd-order, thus making [[media:Pea1.f11.mtg3.djvu|Eqn(1) p.3-3]]a Nonlinear 2nd-order ODE (N2-ODE).

Curvature	$$\chi = \frac{1}{R}$$

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R*1.6: Show $$c_e(Y^1,t)\ddot{Y^1}$$ is nonlinear with respect to. Back to the general L2-ODE-VC [[media:Pea1.f11.mtg4.djvu|Eqn(1) p.4-1.]]