 11.11.1: In each of 1 through 8, compute the requested derivative in two way...
 11.11.13: In each of 1 through 10, a position vector is given. Determine the ...
 11.11.26: In each of 1 through 6, find the streamlines of the vector field an...
 11.11.33: In each of 1 through 6, compute the gradient of the function and ev...
 11.11.43: In each of 1 through 6, compute F and F and verify explicitly that ...
 11.11.2: In each of 1 through 8, compute the requested derivative in two way...
 11.11.14: In each of 1 through 10, a position vector is given. Determine the ...
 11.11.27: In each of 1 through 6, find the streamlines of the vector field an...
 11.11.34: In each of 1 through 6, compute the gradient of the function and ev...
 11.11.44: In each of 1 through 6, compute F and F and verify explicitly that ...
 11.11.3: In each of 1 through 8, compute the requested derivative in two way...
 11.11.15: In each of 1 through 10, a position vector is given. Determine the ...
 11.11.28: In each of 1 through 6, find the streamlines of the vector field an...
 11.11.35: In each of 1 through 6, compute the gradient of the function and ev...
 11.11.45: In each of 1 through 6, compute F and F and verify explicitly that ...
 11.11.4: In each of 1 through 8, compute the requested derivative in two way...
 11.11.16: In each of 1 through 10, a position vector is given. Determine the ...
 11.11.29: In each of 1 through 6, find the streamlines of the vector field an...
 11.11.36: In each of 1 through 6, compute the gradient of the function and ev...
 11.11.46: In each of 1 through 6, compute F and F and verify explicitly that ...
 11.11.5: In each of 1 through 8, compute the requested derivative in two way...
 11.11.17: In each of 1 through 10, a position vector is given. Determine the ...
 11.11.30: In each of 1 through 6, find the streamlines of the vector field an...
 11.11.37: In each of 1 through 6, compute the gradient of the function and ev...
 11.11.47: In each of 1 through 6, compute F and F and verify explicitly that ...
 11.11.6: In each of 1 through 8, compute the requested derivative in two way...
 11.11.18: In each of 1 through 10, a position vector is given. Determine the ...
 11.11.31: In each of 1 through 6, find the streamlines of the vector field an...
 11.11.38: In each of 1 through 6, compute the gradient of the function and ev...
 11.11.48: In each of 1 through 6, compute F and F and verify explicitly that ...
 11.11.7: In each of 1 through 8, compute the requested derivative in two way...
 11.11.19: In each of 1 through 10, a position vector is given. Determine the ...
 11.11.32: Construct a vector field whose streamlines are circles about the or...
 11.11.39: In each of 7 through 10, compute the directional derivative of the ...
 11.11.49: In each of 7 through 12, compute and verify explicitly that () = O(...
 11.11.8: In each of 1 through 8, compute the requested derivative in two way...
 11.11.20: In each of 1 through 10, a position vector is given. Determine the ...
 11.11.40: In each of 7 through 10, compute the directional derivative of the ...
 11.11.50: In each of 7 through 12, compute and verify explicitly that () = O(...
 11.11.9: In each of 9, 10, and 11, (a) write the position vector and tangent...
 11.11.21: In each of 1 through 10, a position vector is given. Determine the ...
 11.11.41: In each of 7 through 10, compute the directional derivative of the ...
 11.11.51: In each of 7 through 12, compute and verify explicitly that () = O(...
 11.11.10: In each of 9, 10, and 11, (a) write the position vector and tangent...
 11.11.22: In each of 1 through 10, a position vector is given. Determine the ...
 11.11.42: In each of 7 through 10, compute the directional derivative of the ...
 11.11.52: In each of 7 through 12, compute and verify explicitly that () = O(...
 11.11.11: In each of 9, 10, and 11, (a) write the position vector and tangent...
 11.11.23: Show that any straight line has curvature zero. Conversely, if a sm...
 11.11.53: In each of 7 through 12, compute and verify explicitly that () = O(...
 11.11.12: Suppose F(t) = x(t)i + y(t)j + z(t)k is the position vector for a p...
 11.11.24: Show that the curvature of a circle is constant. Hint: If the radiu...
 11.11.54: In each of 7 through 12, compute and verify explicitly that () = O(...
 11.11.25: Show that (t) = F (t) F(t) F (t)3 . 11
 11.11.55: Let be a scalar field and F a vector field. Derive expressions for ...
Solutions for Chapter 11: Vector Differential Calculus
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 11: Vector Differential Calculus
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412. Since 55 problems in chapter 11: Vector Differential Calculus have been answered, more than 36336 students have viewed full stepbystep solutions from this chapter. Chapter 11: Vector Differential Calculus includes 55 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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)·(b  Ax) = o.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·