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Solutions for Chapter 9.2: Partial Derivatives

Biocalculus: Calculus for Life Sciences | 1st Edition | ISBN: 9781133109631 | Authors: James Stewart, Troy Day

Full solutions for Biocalculus: Calculus for Life Sciences | 1st Edition

ISBN: 9781133109631

Biocalculus: Calculus for Life Sciences | 1st Edition | ISBN: 9781133109631 | Authors: James Stewart, Troy Day

Solutions for Chapter 9.2: Partial Derivatives

Solutions for Chapter 9.2
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Textbook: Biocalculus: Calculus for Life Sciences
Edition: 1
Author: James Stewart, Troy Day
ISBN: 9781133109631

This textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. Biocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. Since 73 problems in chapter 9.2: Partial Derivatives have been answered, more than 27182 students have viewed full step-by-step solutions from this chapter. Chapter 9.2: Partial Derivatives includes 73 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
  • Augmented matrix [A b].

    Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

  • Characteristic equation det(A - AI) = O.

    The n roots are the eigenvalues of A.

  • Column space C (A) =

    space of all combinations of the columns of A.

  • Complex conjugate

    z = a - ib for any complex number z = a + ib. Then zz = Iz12.

  • Dimension of vector space

    dim(V) = number of vectors in any basis for V.

  • Free columns of A.

    Columns without pivots; these are combinations of earlier columns.

  • Full column rank r = n.

    Independent columns, N(A) = {O}, no free variables.

  • Identity matrix I (or In).

    Diagonal entries = 1, off-diagonal entries = 0.

  • Left inverse A+.

    If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

  • Left nullspace N (AT).

    Nullspace of AT = "left nullspace" of A because y T A = OT.

  • Markov matrix M.

    All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

  • Multiplication Ax

    = Xl (column 1) + ... + xn(column n) = combination of columns.

  • Normal equation AT Ax = ATb.

    Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b - Ax) = o.

  • Outer product uv T

    = column times row = rank one matrix.

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row space.

  • Similar matrices A and B.

    Every B = M-I AM has the same eigenvalues as A.

  • Spectral Theorem A = QAQT.

    Real symmetric A has real A'S and orthonormal q's.

  • Toeplitz matrix.

    Constant down each diagonal = time-invariant (shift-invariant) filter.

  • Transpose matrix AT.

    Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

  • Vector addition.

    v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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