Math 601 Solutions to Homework 10
... where C is any curve on the sphere x2 + y 2 + z 2 = 9. Answer: (a) On the circle C, the value of (x2 + y 2 )2 is 34 = 81. If we integrate this function over the circle, we get 81 times the circumference of the circle, which is 81(3)(2π) = 486π (b) The problem should have specified the orientation of ...
... where C is any curve on the sphere x2 + y 2 + z 2 = 9. Answer: (a) On the circle C, the value of (x2 + y 2 )2 is 34 = 81. If we integrate this function over the circle, we get 81 times the circumference of the circle, which is 81(3)(2π) = 486π (b) The problem should have specified the orientation of ...
math 67a hw 2 solutions
... Claim. Let V be a vector space over F and suppose that W1 , W2 , and W3 are subspace of V such that W1 + W3 = W2 + W3 , then W1 = W2 . Solution The claim is false. Let V = F2 , and W3 = V while W1 = span(0, 1) and W2 = span(1, 0). Then W1 + W3 = V = W2 + W3 but W1 6= W2 . PWE 4.4 Prove or give a cou ...
... Claim. Let V be a vector space over F and suppose that W1 , W2 , and W3 are subspace of V such that W1 + W3 = W2 + W3 , then W1 = W2 . Solution The claim is false. Let V = F2 , and W3 = V while W1 = span(0, 1) and W2 = span(1, 0). Then W1 + W3 = V = W2 + W3 but W1 6= W2 . PWE 4.4 Prove or give a cou ...
In a previous class, we saw that the positive reals R+ is a vector
... continue to use to represent vector addition in R+ , and to represent scalar multiplication. First, notice that if our field F is the real numbers R, our vector space R+ is a one-dimensional vector space, and so must be isomorphic to R with ordinary addition (since they are both finite dimensional ...
... continue to use to represent vector addition in R+ , and to represent scalar multiplication. First, notice that if our field F is the real numbers R, our vector space R+ is a one-dimensional vector space, and so must be isomorphic to R with ordinary addition (since they are both finite dimensional ...
Math 28S Vector Spaces Fall 2011 Definition: Given a field F, a
... F (where the addition and scalar multiplication are done term-by-term). (b) The set F ∞ of infinite sequences where all but finitely many elements of the sequence are 0 also forms a vector space over F , with the same operations. Here, some care needs to be taken to ensure that addition is closed. ( ...
... F (where the addition and scalar multiplication are done term-by-term). (b) The set F ∞ of infinite sequences where all but finitely many elements of the sequence are 0 also forms a vector space over F , with the same operations. Here, some care needs to be taken to ensure that addition is closed. ( ...
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 ...
Section 3.3 Vectors in the Plane
... Vector v in the plane The set of all directed line segments that are equivalent to given directed line segment PQ, written v = PQ. Standard position The representative of a set of equivalent directed line segments whose initial point is the origin. Zero vector A vector whose initial point and termin ...
... Vector v in the plane The set of all directed line segments that are equivalent to given directed line segment PQ, written v = PQ. Standard position The representative of a set of equivalent directed line segments whose initial point is the origin. Zero vector A vector whose initial point and termin ...
Lecture 5 vector bundles, gauge theory tangent bundle In Lecture 2
... where p ∈ M , v ∈ V and g(p) is an invertible linear map on V . This gj←i (p) ∈ GL(V, R) is called a transition function. If there is a triple intersection of three charts Ui , Uj and Uk , the transition function must satisfy the consistency condition, gk←j (p)gj←i (p) = gk←i (p), on p ∈ Ui ∩ Uj ∩ U ...
... where p ∈ M , v ∈ V and g(p) is an invertible linear map on V . This gj←i (p) ∈ GL(V, R) is called a transition function. If there is a triple intersection of three charts Ui , Uj and Uk , the transition function must satisfy the consistency condition, gk←j (p)gj←i (p) = gk←i (p), on p ∈ Ui ∩ Uj ∩ U ...
Document
... Vector spaces in which the scalars are complex numbers are called complex vector spaces, and those in which the scalars must be real are called real vector spaces. ...
... Vector spaces in which the scalars are complex numbers are called complex vector spaces, and those in which the scalars must be real are called real vector spaces. ...
A NOTE ON SUB-BUNDLES OF VECTOR BUNDLES Introduction
... Proposition 1. If F satisfies (*), then F is a sub-bundle if and only if F is the kernel of a vector bundle homomorphism θ : E → E 0 . If F is a sub-bundle, one can take E 0 to be the quotient bundle E/F . For the converse see for example Proposition 1.7.2 in [3]. Proposition 2. If F satisfies (*), ...
... Proposition 1. If F satisfies (*), then F is a sub-bundle if and only if F is the kernel of a vector bundle homomorphism θ : E → E 0 . If F is a sub-bundle, one can take E 0 to be the quotient bundle E/F . For the converse see for example Proposition 1.7.2 in [3]. Proposition 2. If F satisfies (*), ...
Vector field
In vector calculus, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point.The elements of differential and integral calculus extend to vector fields in a natural way. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow).In coordinates, a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law in passing from one coordinate system to the other. Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector). More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field.