Final Exam

... d) If f and g are smooth functions on M , let (f, g) = Ω(Xdg , Xdf ) Show that (df, dg) = d(f, g), and that f is constant along the integral curves of Xdg if and only if g is constant along the integral curves of Xdf . Problem 2: On R4 , with coordinates x1 , x2 , x3 , x4 , consider the vector field ...

... d) If f and g are smooth functions on M , let (f, g) = Ω(Xdg , Xdf ) Show that (df, dg) = d(f, g), and that f is constant along the integral curves of Xdg if and only if g is constant along the integral curves of Xdf . Problem 2: On R4 , with coordinates x1 , x2 , x3 , x4 , consider the vector field ...

Topology/Geometry Aug 2011

... satisfying w2 +x2 +y 2 +z 2 = a and wz −xy = b. y z Show that if a 6= 2|b|, then Mab is a smooth submanifold of R4 . 6. Suppose M is an annulus [a, b] × S 1 , for numbers b > a > 0, with C ∞ Riemannian metric given in polar coordinates r ∈ [a, b] and θ ∈ S 1 by ds2 = dr2 + ϕ(r)2 dθ2 for some functio ...

... satisfying w2 +x2 +y 2 +z 2 = a and wz −xy = b. y z Show that if a 6= 2|b|, then Mab is a smooth submanifold of R4 . 6. Suppose M is an annulus [a, b] × S 1 , for numbers b > a > 0, with C ∞ Riemannian metric given in polar coordinates r ∈ [a, b] and θ ∈ S 1 by ds2 = dr2 + ϕ(r)2 dθ2 for some functio ...

Math 130 Worksheet 2: Linear algebra

... Math 130 Worksheet 2: Linear algebra ... but I thought this was a geometry class! ...

... Math 130 Worksheet 2: Linear algebra ... but I thought this was a geometry class! ...

Topology/Geometry Jan 2012

... Ht (X) = 0 if t > 0 Q.4 Let M be the subset of Euclidean R3 defined by the zeros of the function f (x, y, z) = xy − z. (a) Prove that M is a submanifold of R3 . (b) Define a local coordinate system on M and compute the Riemannian metric induced on M by its embedding into Euclidean R3 in terms of the ...

... Ht (X) = 0 if t > 0 Q.4 Let M be the subset of Euclidean R3 defined by the zeros of the function f (x, y, z) = xy − z. (a) Prove that M is a submanifold of R3 . (b) Define a local coordinate system on M and compute the Riemannian metric induced on M by its embedding into Euclidean R3 in terms of the ...

PDF

... is in [?]. It is also proved in [?] for any prismatoid. The alternate formula is proved in [?]. Some authors impose the condition that the lateral faces must be triangles or trapezoids. However, this condition is unnecessary since it is easily shown to hold. V = ...

... is in [?]. It is also proved in [?] for any prismatoid. The alternate formula is proved in [?]. Some authors impose the condition that the lateral faces must be triangles or trapezoids. However, this condition is unnecessary since it is easily shown to hold. V = ...

MATH M16A: Applied Calculus Course Objectives (COR) • Evaluate

... algebraic techniques and the properties of limits. Determine whether a function is continuous or discontinuous at a point. Calculate the derivative of an algebraic function using the formal definition of the derivative. Explain the concept of derivative as an "instantaneous rate of change" and the s ...

... algebraic techniques and the properties of limits. Determine whether a function is continuous or discontinuous at a point. Calculate the derivative of an algebraic function using the formal definition of the derivative. Explain the concept of derivative as an "instantaneous rate of change" and the s ...

syllabus - The City University of New York

... Hunter College of The City University of New York ...

... Hunter College of The City University of New York ...

Calculus I Midterm II Review Materials Solutions to the practice

... Solutions to the practice problems are provided in another sheet, it’s recommended that you first do the problems by yourself and then check with the answer. 1, Definition of Derivative, Differentials? Relation with change rate, tangent, velocity? Express a limit as √ a derivative and evaluate. ...

... Solutions to the practice problems are provided in another sheet, it’s recommended that you first do the problems by yourself and then check with the answer. 1, Definition of Derivative, Differentials? Relation with change rate, tangent, velocity? Express a limit as √ a derivative and evaluate. ...

1 - eBoard

... the derivative. The only twist is that the coordinate point that is given needs to be inserted in the derivitized formula to find the value of the derivitization at that point. ...

... the derivative. The only twist is that the coordinate point that is given needs to be inserted in the derivitized formula to find the value of the derivitization at that point. ...

Lie groups and Lie algebras 1 Examples of Lie groups

... µ : (g, h) 7→ gh and the inversion map ι : G→G ι : g 7→ g −1 are smooth. ...

... µ : (g, h) 7→ gh and the inversion map ι : G→G ι : g 7→ g −1 are smooth. ...

Ordinary derivative If a is regarded as a vector function of a single

... being taken, and therefore the e1,e2,e3 each has a derivative of identically zero. This often holds true for problems dealing with vector fields in a fixed coordinate system, or for simple problems in physics. However, many complex problems involve the derivative of a vector function in multiple mov ...

... being taken, and therefore the e1,e2,e3 each has a derivative of identically zero. This often holds true for problems dealing with vector fields in a fixed coordinate system, or for simple problems in physics. However, many complex problems involve the derivative of a vector function in multiple mov ...

Solutions - Penn Math

... The domain of F(x, y) is R2 minus the points where x + y = 0, which is not connected since it consists of two pieces, namely the half-planes x + y < 0 and x + y > 0. However, the curve C is contained in the half-plane x + y > 0, and this is simply connected, so we are OK. ...

... The domain of F(x, y) is R2 minus the points where x + y = 0, which is not connected since it consists of two pieces, namely the half-planes x + y < 0 and x + y > 0. However, the curve C is contained in the half-plane x + y > 0, and this is simply connected, so we are OK. ...

Lecture 4

... Let f : R3 → R and g : R3 → R be functions, let v p and w p be tangent vectors to R3 in p, and let a, b ∈ R be numbers. Then (1) (av p + bw p )[ f ] = av p [ f ] + bw p [ f ], (2) v p [a f + bg] = av p [ f ] + bv p [g], (3) v p [ f g] = v p [ f ] · g(p) + f (p) · v p [g]. (1) and (2) mean that v p [ ...

... Let f : R3 → R and g : R3 → R be functions, let v p and w p be tangent vectors to R3 in p, and let a, b ∈ R be numbers. Then (1) (av p + bw p )[ f ] = av p [ f ] + bw p [ f ], (2) v p [a f + bg] = av p [ f ] + bv p [g], (3) v p [ f g] = v p [ f ] · g(p) + f (p) · v p [g]. (1) and (2) mean that v p [ ...

Derivatives and Integrals of Vector Functions

... Solution: (a) According to Theorem 2, we differentiate each component of r: r ′(t) = 3t2i + (1 – t)e–t j + 2 cos 2t k ...

... Solution: (a) According to Theorem 2, we differentiate each component of r: r ′(t) = 3t2i + (1 – t)e–t j + 2 cos 2t k ...

These are some math problems i need today if possible

... n times a year for t years at interest rate of r and initial deposit p is given by the following fromula below/ A=p{1+ r/n)^nt, gind the formula for A if the number of compounding periood per year becomes larger or n∞ ...

... n times a year for t years at interest rate of r and initial deposit p is given by the following fromula below/ A=p{1+ r/n)^nt, gind the formula for A if the number of compounding periood per year becomes larger or n∞ ...

Document

... If the points P and Q have position vectors r(t) and r(t + h), then represents the vector r(t + h) – r(t), which can therefore be regarded as a secant vector. If h > 0, the scalar multiple (1/h)(r(t + h) – r(t)) has the same direction as r(t + h) – r(t). As h 0, it appears that this vector approac ...

... If the points P and Q have position vectors r(t) and r(t + h), then represents the vector r(t + h) – r(t), which can therefore be regarded as a secant vector. If h > 0, the scalar multiple (1/h)(r(t + h) – r(t)) has the same direction as r(t + h) – r(t). As h 0, it appears that this vector approac ...

PHY-2049-003 Physics for Engineers and Scientists

... So the question is how do we get from these last two expressions to the ab sin() concept. Keep in mind that everything here is being done in the x-y plane just for simplicity. If you REALLY want to understand this most important process, follow the rest and then reproduce the calculations in all th ...

... So the question is how do we get from these last two expressions to the ab sin() concept. Keep in mind that everything here is being done in the x-y plane just for simplicity. If you REALLY want to understand this most important process, follow the rest and then reproduce the calculations in all th ...

Ph.D. Qualifying examination in topology Charles Frohman and

... B3) Let Wc = f(x; y; z; w) 2 R4 : xyz = cg and Yc = f(x; y; z; w) 2 R4 : xzw = cg. For what real numbers c is Yc a three-manifold? For what pairs (c1 ; c2 ) is Wc1 \ Yc2 a two-manifold. B4) Consider the one form ...

... B3) Let Wc = f(x; y; z; w) 2 R4 : xyz = cg and Yc = f(x; y; z; w) 2 R4 : xzw = cg. For what real numbers c is Yc a three-manifold? For what pairs (c1 ; c2 ) is Wc1 \ Yc2 a two-manifold. B4) Consider the one form ...

BOOK REVIEW

... A Poisson bracket on a smooth manifold M is bilinear product f, g → {f, g} defined on the space C ∞ (M ) of all smooth functions on M such that the space C ∞ (M ) is a Lie algebra with respect this form and the Leibniz identity {f g, h} = f {g, h}+g{f, h} holds. In this case the manifold M is called ...

... A Poisson bracket on a smooth manifold M is bilinear product f, g → {f, g} defined on the space C ∞ (M ) of all smooth functions on M such that the space C ∞ (M ) is a Lie algebra with respect this form and the Leibniz identity {f g, h} = f {g, h}+g{f, h} holds. In this case the manifold M is called ...

Remarks on the Cartan Formula and Its Applications

... on Riemannian manifolds, complex manifolds and generalized complex manifolds. We would like to point out that such commutator formulas are essentially consequences of the classical Cartan formula for Lie derivative, but they have deep applications in geometry such as in studying the smoothness of de ...

... on Riemannian manifolds, complex manifolds and generalized complex manifolds. We would like to point out that such commutator formulas are essentially consequences of the classical Cartan formula for Lie derivative, but they have deep applications in geometry such as in studying the smoothness of de ...

Derivative of secant

... This is usually written in a diﬀerent fashion; there are often many diﬀerent ways of writing combinations of trigonometric functions. The standard way of writing this is: d sin x ...

... This is usually written in a diﬀerent fashion; there are often many diﬀerent ways of writing combinations of trigonometric functions. The standard way of writing this is: d sin x ...

Course Narrative

... understanding of the concepts and connections among concepts in calculus. The two major themes are the derivative as a rate of change and the integral as an accumulation function. Technology is used throughout the course and students are taught how to use graphing calculators to help solve problems, ...

... understanding of the concepts and connections among concepts in calculus. The two major themes are the derivative as a rate of change and the integral as an accumulation function. Technology is used throughout the course and students are taught how to use graphing calculators to help solve problems, ...

1 Lecture 4 - Integration by parts

... the derivative of f w.r.t x, written df dx literally indicates the infinitesimal change in f compared to the infinitesimal change in x. But the operator “d” actually behaves like a derivative: given f (x) = x2 then df (x) = 2x dx. If we divide both sides by dx, we get df = 2x, dx which is the correc ...

... the derivative of f w.r.t x, written df dx literally indicates the infinitesimal change in f compared to the infinitesimal change in x. But the operator “d” actually behaves like a derivative: given f (x) = x2 then df (x) = 2x dx. If we divide both sides by dx, we get df = 2x, dx which is the correc ...