Indicial Notation, Second Order Tensors, Einstein Summations Let {??} and {?? ′ = ?????} be two...
Indicial Notation, Second Order Tensors, Einstein Summations
Let {??} and {?? ′ = ?????} be two rectangular Cartesian base vectors.
(a) Verify that ?????? = ??? = ??????
(b) Show that if ?? ′ = ????? then ?? = ????? ′ .
Homework Answers
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Indicial Notation, Second Order Tensors, Einstein Summations
Let {??} and {?? ′ = ?????} be
two...
first picture is the notation on the test paper Second picture is the question i would...
first picture is the notation on the test paper
Second picture is the question i would like to solve
thanks~
Notation and convention: r x +y The distance from the origin to the point r [x,y,z] + ê: The unit vector along the direction of r-[x, y,z] (a.e,6)-i.j.):m :The orthonormal bases of a Cartesian coordinate system. for dummy indices Einstein convention: Omitting the summation notation (repeated indices). Examples:ab,-a b, ab a b Notice: No dummy index is allowed to be...
Notation and convention: r x +y The distance from the origin to the point r [x,y,z]...
Notation and convention: r x +y The distance from the origin to the point r [x,y,z] + ê: The unit vector along the direction of r-[x, y,z] (a.e,6)-i.j.):m :The orthonormal bases of a Cartesian coordinate system. for dummy indices Einstein convention: Omitting the summation notation (repeated indices). Examples:ab,-a b, ab a b Notice: No dummy index is allowed to be repeated more than twice. You should change the "names" of the dummy indices before taking the product of two summations...
Parts d and e needed! 1) Here are two rank 1 tensors of differing types, written...
Parts d and e needed!
1) Here are two rank 1 tensors of differing types, written in Cartesian co-ordinates: 1 -4 vi(x, y, z) = wi(x,y,z) 2 3 a) Calculate their scalar product viw. (2 marks) b) Using the appropriate tensor transform equations, and expressions for the partial derivatives obtained from class, transform both tensors into cylindrical polar co-ordinates (r, 0, z). (6 marks) c) Calculate the scalar product of the transformed tensors, and show that the answer is the...
23. Consider the Bessel equation of order zero (a) Show thatr 0 is a double root of the indicial equation. (b) Find one solution of the form 4 22 22.42 22.42.62 This solution is known as Jo(t). (c) F...
23. Consider the Bessel equation of order zero (a) Show thatr 0 is a double root of the indicial equation. (b) Find one solution of the form 4 22 22.42 22.42.62 This solution is known as Jo(t). (c) Find a second solution using the method of reduction of order
23. Consider the Bessel equation of order zero (a) Show thatr 0 is a double root of the indicial equation. (b) Find one solution of the form 4 22 22.42 22.42.62...
Let y-and Г be two alphabets, and let Г be the alphabet be the alphabet of vectors where the first element is from Σ and the second is from「 For example, if -(a,b) and Г-{0.1} then and B c Г be any...
Let y-and Г be two alphabets, and let Г be the alphabet be the alphabet of vectors where the first element is from Σ and the second is from「 For example, if -(a,b) and Г-{0.1} then and B c Г be any regular languages, and consider the language Let A Show A B is regular.
Let y-and Г be two alphabets, and let Г be the alphabet be the alphabet of vectors where the first element is from Σ and...
4. Let f = ez?y. (a) Find the second-order Taylor's polynomial of f about (1,0). (b)...
4. Let f = ez?y. (a) Find the second-order Taylor's polynomial of f about (1,0). (b) Find and classify the critical points of f. (c) Consider two constant vectors A, B E R3 and a scalar field g: R3 → R. Define h : R + R as h(t) = g(A + tB), for every t E R. Find suitable formulas for C0, C1, C2 depending on g, A, B such that h(t) = co + ci(t – 1) +c2(t...
1. Let L: P1(R) + P1(R) be a linear transformation given by L(a + bx) =...
1. Let L: P1(R) + P1(R) be a linear transformation given by L(a + bx) = a - b + (2a – b)x. Let S = {1, 2} and T = {1+x} be two basis for P1(R). (a) Find the matrix A of L with respect to basis S. (a) Find the matrix B of L with respect to basis T. (c) Find the matrix P obtained by expressing vectors in basis T in terms of vectors in basis (d)...
I. Let f : R2 → R be defined by f(x)l cos (122) 211 Compute the second order Taylor polynomial of...
I. Let f : R2 → R be defined by f(x)l cos (122) 211 Compute the second order Taylor polynomial of f near the point xo - 0. A Road Map to Glory (On your way to glory, please keep in mind that f is class C) a) Fill in the blanks: The second order Taylor's polynomial at h E R2 is given by T2 (h) = 2! b) Compute the numbers, vectors and matrices that went into the blanks...
Let two times differentiable in the point . The first and second order differentiable equation of...
Let
two times differentiable in the point
. The first and second order differentiable equation of in , imply that
the functions
and
, given by:
satisfies
and
.
Prove that if
with
then it satisfy
f: RR a ER f We were unable to transcribe this imageተ ፖ : R Ꭱ : ] . r(h) = f(a+h)-f(a) – f'ah R(t) = f'(a +t) - f'(a) - f"(a)t r(h) lim h 0 h -0 lim R(t) h 0 u:R u(w)...
7. Let A be a 4 x 3 matrix, and let b and y be two...
7. Let A be a 4 x 3 matrix, and let b and y be two arbitrary vectors in R. We are told that the system Ax- b has a unique solution. What can you say about the number of solutions of the system Ax - y? Explain your answer. 8. Let u. v, w, b be arbitrary vectors in R". Suppose that b = x1u+xy+23w for some scalars i, r23. Show that Span u, v, w, b Span u,...
first picture is the notation on the test paper
Second picture is the question i would like to solve
thanks~
Notation and convention: r x +y The distance from the origin to the point r [x,y,z] + ê: The unit vector along the direction of r-[x, y,z] (a.e,6)-i.j.):m :The orthonormal bases of a Cartesian coordinate system. for dummy indices Einstein convention: Omitting the summation notation (repeated indices). Examples:ab,-a b, ab a b Notice: No dummy index is allowed to be...
Notation and convention: r x +y The distance from the origin to the point r [x,y,z] + ê: The unit vector along the direction of r-[x, y,z] (a.e,6)-i.j.):m :The orthonormal bases of a Cartesian coordinate system. for dummy indices Einstein convention: Omitting the summation notation (repeated indices). Examples:ab,-a b, ab a b Notice: No dummy index is allowed to be repeated more than twice. You should change the "names" of the dummy indices before taking the product of two summations...
Parts d and e needed!
1) Here are two rank 1 tensors of differing types, written in Cartesian co-ordinates: 1 -4 vi(x, y, z) = wi(x,y,z) 2 3 a) Calculate their scalar product viw. (2 marks) b) Using the appropriate tensor transform equations, and expressions for the partial derivatives obtained from class, transform both tensors into cylindrical polar co-ordinates (r, 0, z). (6 marks) c) Calculate the scalar product of the transformed tensors, and show that the answer is the...
23. Consider the Bessel equation of order zero (a) Show thatr 0 is a double root of the indicial equation. (b) Find one solution of the form 4 22 22.42 22.42.62 This solution is known as Jo(t). (c) Find a second solution using the method of reduction of order
23. Consider the Bessel equation of order zero (a) Show thatr 0 is a double root of the indicial equation. (b) Find one solution of the form 4 22 22.42 22.42.62...
Let y-and Г be two alphabets, and let Г be the alphabet be the alphabet of vectors where the first element is from Σ and the second is from「 For example, if -(a,b) and Г-{0.1} then and B c Г be any regular languages, and consider the language Let A Show A B is regular.
Let y-and Г be two alphabets, and let Г be the alphabet be the alphabet of vectors where the first element is from Σ and...
4. Let f = ez?y. (a) Find the second-order Taylor's polynomial of f about (1,0). (b) Find and classify the critical points of f. (c) Consider two constant vectors A, B E R3 and a scalar field g: R3 → R. Define h : R + R as h(t) = g(A + tB), for every t E R. Find suitable formulas for C0, C1, C2 depending on g, A, B such that h(t) = co + ci(t – 1) +c2(t...
1. Let L: P1(R) + P1(R) be a linear transformation given by L(a + bx) = a - b + (2a – b)x. Let S = {1, 2} and T = {1+x} be two basis for P1(R). (a) Find the matrix A of L with respect to basis S. (a) Find the matrix B of L with respect to basis T. (c) Find the matrix P obtained by expressing vectors in basis T in terms of vectors in basis (d)...
I. Let f : R2 → R be defined by f(x)l cos (122) 211 Compute the second order Taylor polynomial of f near the point xo - 0. A Road Map to Glory (On your way to glory, please keep in mind that f is class C) a) Fill in the blanks: The second order Taylor's polynomial at h E R2 is given by T2 (h) = 2! b) Compute the numbers, vectors and matrices that went into the blanks...
Let
two times differentiable in the point
. The first and second order differentiable equation of in , imply that
the functions
and
, given by:
satisfies
and
.
Prove that if
with
then it satisfy
f: RR a ER f We were unable to transcribe this imageተ ፖ : R Ꭱ : ] . r(h) = f(a+h)-f(a) – f'ah R(t) = f'(a +t) - f'(a) - f"(a)t r(h) lim h 0 h -0 lim R(t) h 0 u:R u(w)...
7. Let A be a 4 x 3 matrix, and let b and y be two arbitrary vectors in R. We are told that the system Ax- b has a unique solution. What can you say about the number of solutions of the system Ax - y? Explain your answer. 8. Let u. v, w, b be arbitrary vectors in R". Suppose that b = x1u+xy+23w for some scalars i, r23. Show that Span u, v, w, b Span u,...
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