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 [ ...
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. ...
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 ...
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. ...
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. ...
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. ...
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 ...
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 ...
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 = ...
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 ...
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 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 ...
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 ...