Sample Final Exam
... Find a basis for this subspace. Answer: Suppose that p(x) = ax2 + bx + c is a polynomial in S. Then, p(2) = 4a + 2b + c and p(1) = a + b + c, so that p(2) − p(1) = 3a + b. Thus, 3a + b = 0, so b = −3a. Thus, we can write p(x) as p(x) = ax2 − 3ax + c = a(x2 − 3x) + c Thus, every polynomial in S is in ...
... Find a basis for this subspace. Answer: Suppose that p(x) = ax2 + bx + c is a polynomial in S. Then, p(2) = 4a + 2b + c and p(1) = a + b + c, so that p(2) − p(1) = 3a + b. Thus, 3a + b = 0, so b = −3a. Thus, we can write p(x) as p(x) = ax2 − 3ax + c = a(x2 − 3x) + c Thus, every polynomial in S is in ...
A QUASI-LINEAR VISCOELASTIC RHEOLOGICAL MODEL FOR
... stress-strain relation in the elastic region of thermoplastics and thermohardening polymers (05%), cf Suchocki (2011). For that purpose several researchers have proposed various, alternative stored-energy potentials. Murnagham material model has been used to capture the nonlinear elasticity for smal ...
... stress-strain relation in the elastic region of thermoplastics and thermohardening polymers (05%), cf Suchocki (2011). For that purpose several researchers have proposed various, alternative stored-energy potentials. Murnagham material model has been used to capture the nonlinear elasticity for smal ...
Momentum and Collision Notes
... momentum is decreased by same impulse – the same products of force and time. However, impact force is greater into the wall than it is into the haystack as the haystack extends impact time, lessening the impact force. Impact time is the time during which momentum is brought to zero. ...
... momentum is decreased by same impulse – the same products of force and time. However, impact force is greater into the wall than it is into the haystack as the haystack extends impact time, lessening the impact force. Impact time is the time during which momentum is brought to zero. ...
Lecture 1 - Lie Groups and the Maurer-Cartan equation
... algebra of left-invariant vector fields on the manifold G. Since this is a Lie subalgebra of the Lie algebra of all differentiable vector fields under the bracket, the Jacobi identity and antisymmetry hold, so we have a lie algebra g canonically associated with the group G, with dim g = dim G. We h ...
... algebra of left-invariant vector fields on the manifold G. Since this is a Lie subalgebra of the Lie algebra of all differentiable vector fields under the bracket, the Jacobi identity and antisymmetry hold, so we have a lie algebra g canonically associated with the group G, with dim g = dim G. We h ...
Problem set 3
... (5) Let F : Fn → Fm be a linear transformation. (a) Prove that if n < m then F is not surjective. (Hint: take a basis for Fn , apply F to it. Explain why the resulting vectors can’t span W . Explain why this implies F is not surjective.) (b) Let F : Fn → Fm be a linear transformation. Prove that if ...
... (5) Let F : Fn → Fm be a linear transformation. (a) Prove that if n < m then F is not surjective. (Hint: take a basis for Fn , apply F to it. Explain why the resulting vectors can’t span W . Explain why this implies F is not surjective.) (b) Let F : Fn → Fm be a linear transformation. Prove that if ...
Collisions etc
... The questions on rotational motion on SAT II Physics deal only with rigid bodies. A rigid body is an object that retains its overall shape, meaning that the particles that make up the rigid body stay in the same position relative to one another. A pool ball is one example of a rigid body since the s ...
... The questions on rotational motion on SAT II Physics deal only with rigid bodies. A rigid body is an object that retains its overall shape, meaning that the particles that make up the rigid body stay in the same position relative to one another. A pool ball is one example of a rigid body since the s ...
Notes on the Dual Space Let V be a vector space over a field F. The
... There is a canonical mapping R of a vector space V into its second dual V ∗∗ = (V ∗ )∗ defined by R(v) = v ∗∗ where v ∗∗ (φ) = φ(v). The proof of the linearity of v ∗∗ and R are left to the reader. If R(v) = 0 we have φ(v) = 0 for all φ ∈ V ∗ . If v 6= 0 then it can be completed to a basis B of V . ...
... There is a canonical mapping R of a vector space V into its second dual V ∗∗ = (V ∗ )∗ defined by R(v) = v ∗∗ where v ∗∗ (φ) = φ(v). The proof of the linearity of v ∗∗ and R are left to the reader. If R(v) = 0 we have φ(v) = 0 for all φ ∈ V ∗ . If v 6= 0 then it can be completed to a basis B of V . ...
03 Spherical Geometry
... The second can be proven by working with the tangents instead of the great circles. Let sides a and b be equal. Extent the radius through point C and draw the tangents to great circles AC and BC in space. They will both hit this extended radius, (call this point S) and since they are two tangents to ...
... The second can be proven by working with the tangents instead of the great circles. Let sides a and b be equal. Extent the radius through point C and draw the tangents to great circles AC and BC in space. They will both hit this extended radius, (call this point S) and since they are two tangents to ...
Torque Torque is defined as the measure of tendency of a force to
... Law of Uniform Angular Motion Consider that you are at the batting cage and about to step into the cage to hit a ball. When selecting a bat, what do you consider? The weight of the bat? Length? Looks? Considering the first two parameters, what do they have to do with swinging the bat? Why do we tel ...
... Law of Uniform Angular Motion Consider that you are at the batting cage and about to step into the cage to hit a ball. When selecting a bat, what do you consider? The weight of the bat? Length? Looks? Considering the first two parameters, what do they have to do with swinging the bat? Why do we tel ...