## Thermodynamics

Published in: Physics
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• ### Sidhant S

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• 1
Integration Formulas: Integration by parts: du = In lul a au du In a cos u du = sin u sin u du cos u du du 2 2 du Formulas from Trigonometry: Sin2 —I— cos2 — I sin(A ± B) — sin A cos B ± cos A sin B tan A±tan B cos(A ± B) — cos A cos B sin A sin B tan(A ± B) 1 *tan A tan B cos 2A — cos2 A — sin2 A sin 2A — 2 sin A cos A 2 tan A tan 2A — I—tan2 A COS — ± 1+cosA sin2 A — 1 1 cos 2A 5 5 Sin —I— Sin — 2 Sin —I— B) cos — B) cos + cos B — 2 cos (A —I— B) cos — B) sin A sin B — — B) — cos(A + B)} Sin COSB — — B) —I— Sin (A + tan u du — — In cos u sin — tan — 2 cos2 A — 1 —cos A 2 sin A I —F COS — + COS sin A — sin B — 2 cos + B) sin — B) cosA — cosB — 2 sin + B) sin — A) cos cos B — 3 {COS(A — B) + COS(A —I— B) } Differentiation Formulas: dy dy du Chain rule: da; d — sin u a— tan u da; d dc v (du //dx) —u (dv/dx) 2 du — cos u— dc — sec2 udu — cos u sin— u — sin u— dc du 1 da; < sin— 5 — < tan 2 — cos— u tan- u — 1 du 35) v du u du da; In u — 1 du u dc dc — 1 du — (0 < cos log e du u da; u dv uv u u 1 sin 2u sin 2u 4 In (u — sin u cos u = I (u + sin u cos u) In(u+ u2 + (12) u2 + a 2 eac (a sin bx—b cos bx) eac sin bx dx euclu = eu sin u du — cos u tan u du = tan u du 1 —1 = - tan 2 a a du sin—I 2 a du 2 u a a sin aa; x sin ax dx a 02 + b2 x cos aa; a 2 a; cos ax a a; sin ax dx sin 2aa; 2 a; a 2 a sin ax eac cos bx dx x sin ax (Ix cos ax dc cos ax dx = cos —k x cos ax dx tan aa; tan ax dc a In x dx = x Inx — x xeax dx x In x dx b) — (Inx eac (a cos bx+b sin ba;) 02 + b2 = Sin —k 2 a a; sin ax cos ax a a sin 2aa; l) 2 2 1
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Summation Formulas: — at N2 k=Nl kak kak I—a a a {I — (n I)an —I— 1 I—a I—a Signals: ö(t) — at Xl(t), X2(t)) — u[n] — u[n — 1] (t)x; (t) dt Complex Exponential Signals: j wot e Distinct signals for distinct wo Periodic for any choice of wo Fundamental frequency wo Fundamental period: wo = 0: undefined Systems: jwo n e Identical signals for values of wo separated by multiples of '27T Periodic only if Wo/(27T) = m/ N G Q Fundamental frequency wo/m Fundamental period: wo = 0: one System H is linear if H {axl (t) + bX2(t)} — afl{X1 + System H is time invariant if — to)} — System H is memoryless if the current output does not depend on future or past inputs. System H is invertible if distinct inputs produce distinct outputs. System H is invertible if an inverse system G exists which "undoes" the action of H. System H is causal if the current output does not depend on future inputs. LTI system H is causal iff h(t) — 0 V t < 0. Bounded: x(t) is bounded if 3 B e R, B > 0, such that x(t) B V t e IR. System H is BIBO stable if every bounded input produces a bounded output. h(t) dt < 00. LTI system H is BIBO stable iff y(t) x(t) * h(t) k = —00 s(t) — T) dT k = —00 t h(T) dT h(t) h[k] h[n] k = —00 2 — T)h(T) dT s(t) —sn —sn— 1]

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