Part I Vectors and Linear Functions

1 Multi-variable Vector-valued Functions (review) 3
1.1 Vectors and inner product 3
1.2 Domain, range, and codomain 7
1.3 Graphs, images, level sets, and contours 10

2 Matrix Multiplication and Determinant 15
2.1 Notation 15
2.2 Matrix multiplication 17

3 Matrix Multiplication and Linear Functions 23
3.1 Matrix multiplication (continued) 23

4 Eigenvalues and Composition of Linear Functions 31
4.1 Composition of linear functions 31
4.2 Eigenvalues and eigenvectors 35

5 Parallelotopes and Volume Scaling Factor 41
5.1 Cross product 41
5.2 Cubes and parallelotopes 44
5.3 Volume scaling factor and parallelotopes 46

Part II Derivative of Multi-variable Functions

6 Continuity of Multi-variable Functions 55
6.1 Limit and continuity in Rn 55
6.2 Discontinuous multi-variable functions 62
6.3 Composition of two functions 66

7 Directional and Partial Derivatives 69
7.1 Directional derivative 69
7.2 Partial derivative 72
7.3 Partial derivatives with constrained variables 75
7.4 Change of variables 78

8 Differentiability 83
8.1 Derivative matrix and gradient vector 83
8.2 Differentiability and derivatives 86
9 The Chain Rule 95
9.1 The chain rule 95
9.2 Zero-level set and graph of derivatives 100

10 Line Integral 109
10.1 Parametrized curves and chain rule 109
10.2 Directional derivative and chain rule 111
10.3 Line integral 114

11 Extreme Values 121
11.1 Extreme values and the Hessian matrix 121
11.2 Criterion for maximum, minimum, and saddle 124
11.3 Lagrange multiplier 129

12 Taylor’s Formula for Multi-Variable Functions 135
12.1 Taylor’s formula for one-variable functions 135
12.2 Higher order directional derivatives 137
12.3 Taylor’s formula for n-variable functions 139

Part III Integration of Multi-variable Functions

13 Double and Triple Integration on Rectangular Domains 147
13.1 Riemann integration in (a review) 147
13.2 Riemann integration in 149
13.3 Riemann integration in 152
13.4 Iterated integrals 153

14 Integration over General Domains 159
14.1 Two types of domains 160
14.2 Double integration 161
14.3 Triple integration 165

15 Integration with Variable Changes 171
15.1 Volume scaling factor 171
15.2 Linear approximation 174

16 Coordinate Systems 181
16.1 Variable change for multiple integrals 181
16.2 Polar coordinates 184
17 Cylindrical and Spherical Coordinates 191
17.1 Cylindrical coordinates 191
17.2 Spherical coordinates 195
18 Surface Integral 199
18.1 Functions on parameterized surfaces 200
18.2 Area scaling factor when 202
18.3 Surface integral 203

Part IV Integration of Vector Fields

19 Line Integral for Tangential Component 211
19.1 Line integral for a scalar function 211
19.2 Line integral for a force field 213
19.3 Path independence, potential, and conservative fields 217

20 Potential Field versus Fluid Flow 221
20.1 Potential field is conservative 221
20.2 Line integral and closed curves 223
20.3 Flow and circulation 226

21 Surface Integral for Normal Components 231
21.1 Surface integral for a scalar function 231
21.2 Surface, normal vector, and tangent plane 233
21.3 Surface integral for a vector field 235

22 Divergence Theorem 241
22.1 Boundary of curves and surfaces 241
22.2 Divergence and the divergence theorem 243
22.3 Divergence theorem in 249
23 Divergence Theorem and Conservation Laws 253
23.1 Flux and conservation laws 253
23.2 Mass conservation 256
23.3 Gauss law with concentrated charge density 259

24 Stokes’ Theorem 263
24.1 Curl of a vector field 263
24.2 Stokes’ theorem 265

25 Stokes’ Theorem and Applications 275
25.1 Simply connected domain 275
25.2 Examples 276
25.3 Faraday’s Law of Electromagnetic Induction 279
25.4 Maxwell Equations 280

Part V Appendix

26 Miscellaneous 287
26.1 Moments and Center of Mass 287
26.2 Green’s Theorem 289
26.3 The Speed of Light and the Problem of Coordinate Systems 290
27 Heat Equation and Diffusion Equation 293
27.1 Homogeneous Heat Equation 293
27.2 Nonlinear Heat Equation 296
27.3 Heat Equation in a Heterogeneous Environment 297
27.4 Random Walk and Diffusion Equation 299
Index 303