Vector Fields Practice Exam Solutions

Calculus 3

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Vectors & 3D Space — Topic Review (⏱️ < 79 min)

Study Guide (fill this in as you watch the videos!)
3D Coordinates, Distance, & Spheres (p. 3-6) (14:23)
Vectors & Basic Operations (p. 7-13) (16:55)
The Dot Product & Vector Projections (p. 14-17) (13:35)
Direction Angles & Direction Cosines (p. 18-19) (4:17)
The Cross Product (p. 20-25) (25:30)
Applications of the Cross Product: Torque (p. 26) (3:38)

Vectors & 3D Space — Practice Exam Questions

Practice Exam Questions
Practice Exam Questions Answer Key
3D Coordinates, Distance & Spheres Practice Exam Solutions (2:47)
Vectors & Basic Operations Practice Exam Solutions (5:55)
The Dot Product & Vector Projections Practice Exam Solutions (19:22)
The Cross Product (10:21)

Lines, Planes & Surfaces — Topic Review (⏱️ < 85 min)

Study Guide (fill this in as you watch the videos!)
Lines in 3D (p. 3-6) (16:46)
Planes in 3D (p. 7-9) (16:39)
Finding the Line of Intersection Between Planes (p. 10-12) (10:53)
Distance Involving Points, Lines, & Planes in 3D (p. 13-15) (10:53)
Cylinders & Quadric Surfaces (p. 16-27) (28:59)

Lines, Planes & Surfaces — Practice Exam Questions

Practice Exam Questions
Practice Exam Questions Answer Key
Lines in 3D Practice Exam Solutions (14:11)
Planes in 3D Practice Exam Solutions (25:15)
Cylinders & Quadric Surfaces Practice Exam Solutions (24:04)

Vector Functions — Topic Review (⏱️ < 77 min)

Study Guide (fill this in as you watch the videos!)
Vector Functions & Space Curves (p. 3-7) (15:26)
Derivatives & Integrals of Vector Functions (p. 8-11) (13:50)
Arc Length & Curvature (p. 12-14) (13:59)
Normal & Binormal Vectors (p. 15-17) (16:10)
The TNB Frame & Torsion (p. 18-19) (8:21)
Velocity & Acceleration in 3D Space (p. 20-22) (9:11)

Vector Functions — Practice Exam Questions

Practice Exam Questions
Practice Exam Questions Answer Key
Vector Functions & Space Curves Practice Exam Solutions (5:28)
Derivatives & Integrals of Vector Functions Practice Exam Solutions (14:16)
Arc Length & Curvature Practice Exam Solutions (38:41)
Normal & Binormal Vectors Practice Exam Solutions (20:55)
Velocity & Acceleration Practice Exam Solutions (9:30)

Multivariate Functions & Partial Derivatives — Topic Review (⏱️ < 92 min)

Study Guide (fill this in as you watch the videos!)
Functions of More Than One Variable (p. 3-7) (16:09)
Limits & Continuity for Multivariate Functions (p. 8-10) (11:45)
Partial Derivatives (p. 11-15) (10:17)
Contour Maps & Partial Derivatives (p. 16-17) (12:35)
Tangent Planes, Linear Approximations & Differentials (p. 18-22) (16:02)
The Chain Rule in Multiple Variables (p. 23-25) (16:39)
Implicit Differentiation for Multivariable Functions (p. 26-29) (8:24)

Multivariate Functions & Partial Derivatives — Practice Exam Questions

Practice Exam Questions
Practice Exam Questions Answer Key
Functions of More Than One Variable Practice Exam Solutions (8:16)
Limits & Continuity for Multivariate Functions Practice Exam Solutions (11:46)
Partial Derivatives Practice Exam Solutions (11:48)
Tangent Planes, Linear Approximations, & Differentials Practice Exam Solutions (16:02)
The Chain Rule in Multiple Variables Practice Exam Solutions (19:38)
Implicit Differentiation Practice Exam Solutions (8:56)

The Gradient, Directional Derivative, & Second Derivative Test — Topic Review (⏱️ < 43 min)

Study Guide (fill this in as you watch the videos!)
The Gradient Vector & Directional Derivatives (p. 3-6) (19:45)
The Gradient & Tangent Planes to Level Surfaces (p. 7-8) (5:23)
Local Extrema & the Second Derivative Test (p. 9-12) (17:00)

The Gradient, Directional Derivative, & Second Derivative Test — Practice Exam Questions

Practice Exam Questions
Practice Exam Questions Answer Key
The Gradient Vector & Directional Derivative Practice Exam Solutions (27:02)
Extreme Values & the Second Derivative Test Practice Exam Solutions (16:52)

Absolute Extrema & Lagrange Multipliers — Topic Review (⏱️ < 64 min)

Study Guide (fill this in as you watch the videos!)
Absolute Minimums & Maximums (p. 3-7) (22:29)
Lagrange Multipliers (p. 8-11) (24:16)
Lagrange Multipliers - Applied Problem Types (p. 12-14) (17:14)

Absolute Extrema & Lagrange Multipliers — Practice Exam Questions

Practice Exam Questions
Practice Exam Questions Answer Key
Absolute Minimums & Maximums Practice Exam Solutions (51:11)
Lagrange Multipliers Practice Exam Solutions (24:25)

Double Integrals — Topic Review (⏱️ < 83 min)

Study Guide (fill this in as you watch the videos!)
Double Integrals, Volume, & Iterated Integrals (p. 3-6) (15:00)
Double Integrals Over General Regions (p. 7-10) (22:10)
Average Value (p. 11) (6:24)
Changing the Order of Integration (p. 12-14) (15:35)
Double Integrals in Polar Coordinates (p. 15-18) (23:56)
Parent Graphs Reference

Double Integrals — Practice Exam Questions

Practice Exam Questions
Practice Exam Questions Answer Key
Double Integrals Mixed Practice Practice Exam Solutions (80:43)

Triple Integrals & Change of Variables — Topic Review (⏱️ < 106 min)

Study Guide (fill this in as you watch the videos!)
Triple Integrals (p. 3-6) (24:33)
Changing the Order of Integration (p. 7-9) (17:48)
Triple Integrals in Cylindrical Coordinates (p. 10-13) (16:48)
Triple Integrals in Spherical Coordinates (p. 14-20) (28:55)
Change of Variables & the Jacobian (p. 21-24) (17:49)

Triple Integrals & Change of Variables — Practice Exam Questions

Practice Exam Questions
Practice Exam Questions Answer Key
Changing the Order of Integration Practice Exam Solutions (19:34)
Cylindrical Coordinates Practice Exam Solutions (15:45)
Spherical Coordinates Practice Exam Solutions (33:05)
Change of Variables Practice Exam Solutions (20:30)
Triple Integrals Mixed Practice Practice Exam Solutions (51:58)

Applications of Multiple Integrals — Topic Review (⏱️ < 52 min)

Study Guide (fill this in as you watch the videos!)
Density & Mass (p. 3-4) (8:00)
Moments & Centers of Mass (p. 5-7) (14:19)
Moment of Inertia (p. 8-9) (7:55)
Probability (p. 10-11) (9:13)
Surface Area (p. 12-14) (14:15)

Applications of Multiple Integrals — Practice Exam Questions

Practice Exam Questions
Practice Exam Questions Answer Key
Center of Mass & Moment of Inertia Practice Exam Solutions (16:57)
Probability Practice Exam Solutions (7:42)
Surface Area Practice Exam Solutions (12:51)

Vector Fields, Line Integrals, & Green's Theorem — Topic Review (⏱️ < 129 min)

Study Guide (fill this in as you watch the videos!)
Vector Fields (p. 3-9) (16:48)
Line Integrals (p. 10-17) (26:56)
Line Integrals of Vector Fields (p. 18-21) (9:51)
The Fundamental Theorem for Line Integrals (p. 22-27) (16:51)
Conservative Vector Fields & Potential Functions (p. 28-31) (13:59)
3D Vector Fields & Potential Functions (p. 32-34) (13:14)
Green's Theorem (p. 35-40) (21:31)
Extending Green's Theorem to Other Region Types (p. 41-44) (9:27)

Vector Fields, Line Integrals, & Green's Theorem — Practice Exam Questions

Practice Exam Questions
Practice Exam Questions Answer Key
Vector Fields Practice Exam Solutions (9:49)
Line Integrals Practice Exam Solutions (22:01)
The Fundamental Theorem for Line Integrals Practice Exam Solutions (28:43)
Green's Theorem Practice Exam Solutions (19:40)

Curl, Divergence, & Surface Integrals — Topic Review (⏱️ < 132 min)

Study Guide (fill this in as you watch the videos!)
Curl (p. 3-5) (8:58)
Divergence (p. 6-7) (4:23)
Vector Forms of Green's Theorem (p. 8-10) (11:02)
Parametric Surfaces (p. 11-18) (18:14)
Tangent Planes & Parametric Surfaces (p. 19-20) (7:25)
Surface Area (p. 21-24) (21:51)
Surface Integrals (p. 25-29) (31:36)
Surface Integrals of Vector Fields (Flux) (p. 30-36) (28:05)

Curl, Divergence, & Surface Integrals — Practice Exam Questions

Practice Exam Questions
Practice Exam Questions Answer Key
Curl & Divergence Practice Exam Solutions (7:19)
Parametric Surfaces Practice Exam Solutions (20:37)
Surface Area Practice Exam Solutions (26:19)
Surface Integrals & Flux Practice Exam Solutions (38:20)

Stokes' Theorem & The Divergence Theorem — Topic Review (⏱️ < 31 min)

Study Guide (fill this in as you watch the videos!)
Stokes' Theorem & Circulation (p. 3-5) (8:25)
The Divergence Theorem & Flux (p.6-8) (14:01)
Summary of Equations for Vector Calculus (p. 9-11) (7:41)

Stokes' Theorem & The Divergence Theorem — Practice Exam Questions

Practice Exam Questions
Practice Exam Questions Answer Key
Stokes' Theorem (23:47)
The Divergence Theorem (6:22)
Vector Calculus Mixed Practice (34:42)

Parametric Equations — Topic Review (⏱️ < 40 min)

Study Guide (fill this in as you watch the videos!)
Parametric Equations Overview (p. 3-6) (13:12)
The First Derivative with Parametric Equations (p. 7-8) (8:19)
The Second Derivative with Parametric Equations (p. 9) (6:53)
Area with Parametric Equations (p. 10) (5:24)
Arc Length with Parametric Equations (p. 11) (5:57)

Parametric Equations — Practice Exam Questions

Practice Exam Questions
Practice Exam Questions Answer Key
Graphing Parametric Equations Practice Exam Solutions (5:30)
The First Derivative with Parametric Equations Practice Exam Solutions (8:08)
The Second Derivative with Parametric Equations Practice Exam Solutions (5:23)
Area & Arc Length with Parametric Equations Practice Exam Solutions (12:18)

Polar Coordinates — Topic Review (⏱️ < 77 min)

How to Go Through This Study Guide (1:01)
Study Guide (fill this in as you watch the videos!)
Polar Coordinates Integral and Area Calculator
Polar Coordinates Overview (p. 3-4) (6:53)
Converting Between Polar and Rectangular Coordinates (p. 5-6) (5:56)
Converting Between Polar and Rectangular Equations (p. 7-8) (4:58)
Polar Equations and Graphs (p. 9-12) (5:14)
Graphing Circles in Polar Form (p. 13) (2:14)
Graphing Cardioids and Limaçons (p. 14-17) (11:16)
Graphing Roses (p. 18-19) (4:32)
Area of Polar Curves (p. 20-21) (5:54)
Area Between Polar Curves (p. 22-24) (23:42)
Derivatives Involving Polar Curves (p. 25) (5:27)

Polar Coordinates — Practice Exam Questions

Practice Exam Questions
Practice Exam Questions Answer Key
Converting Between Polar and Rectangular Coordinates Practice Exam Solutions (4:36)
Areas Involving Polar Curves Practice Exam Solutions (43:01)
Derivatives Involving Polar Curves Practice Exam Solutions (3:56)

[OPTIONAL] Sh*t You Forgot from Calc 2

Study Guide (print me!)
Improper Integrals (p. 3-6) (16:42)
Deciding the Right Integration Method to Use (p. 7-8) (7:55)
Integration by Parts Overview (p. 9-11) (7:50)
IBP - Algebraic and Trig/Exponential (ATE) (p. 12-13) (5:27)
IBP - Algebraic and Log/Inverse Trig (LIA) (p. 14-15) (6:50)
IBP - Trig and Exponential (Circular Integration) (p. 16-17) (10:44)
Substitution and IBP in the Same Problem (p. 18) (4:20)
Trig Integrals Overview (p. 19-20) (5:04)
Trig Integrals with Powers of Sine and Cosine (p. 21-23) (15:53)
Trig Integrals with Sine and Cosine with Different Angles (p. 24) (3:28)
Trig Integrals with Tangent and Secant - TOES (p. 25-27) (11:26)
Trig Integrals with Tangent and Secant - No TOES (p. 28-29) (11:46)
Partial Fraction Decomposition - Overview (p. 30-32) (11:59)
Integration with Linear Factors (p. 33-34) (13:58)
Integration with Irreducible Quadratic Factors (p. 35-36) (12:23)

[OPTIONAL] Sh*t You Forgot from Calc 1

Study Guide (print me!)
Limits - Overview (p. 3-6) (11:20)
Evaluating Limits - Overview (p. 7) (5:36)
Evaluating Limits Algebraically - Piecewise Functions (p. 8) (5:26)
Evaluating Limits Algebraically - 0/0 Form (p. 9-12) (21:28)
Evaluating Infinite Limits - Number/0 Form (p. 13) (5:56)
Evaluating Infinite Limits - Knowing Graphs (p. 14-15) (5:18)
The Squeeze Theorem (p. 16) (4:42)
Limits at Infinity and Horizontal Asymptotes (p. 17-23) (23:09)
Derivative Rules - Overview (p. 24-25) (7:52)
Product and Quotient Rules (p. 26-27) (12:35)
The Chain Rule (p. 28-29) (12:27)
Indeterminate Forms and L'Hôpital's Rule (p. 30-32) (30:04)
Growth Rates (How to Skip L'Hôpital's Rule) (p. 33) (4:51)
How to Do Nasty Limits and Skip the Algebra 😈 (p. 34) (4:10)
Antiderivatives and Indefinite Integrals (p. 35-38) (13:38)
Substitution (p. 39-42) (16:26)
My F****ing Variable Didn't Cancel 😵‍💫🤬😭 (p. 43) (4:23)
Substitution with Definite Integrals (p. 44-46) (11:52)
Substitution vs Basic Integration (p. 47-48) (10:54)

[OPTIONAL] Sh*t You Forgot from Trig

Study Guide (print me!)
Trig Functions (p. 3) (4:25)
The Unit Circle (p. 4-7) (17:35)
Trig Identities (p. 8-9) (7:39)
Solving Trig Equations and Inequalities (p. 10-11) (16:13)
Inverse Trig Overview (p. 12-13) (7:46)
Evaluating Mixed Trig Function Expressions (p. 14-17) (10:47)
Evaluating Mixed Trig Function Expressions with Variables (p. 18) (3:50)

[OPTIONAL] Sh*t You Forgot from Precalc

Study Guide (print me!)
Special Products and Factoring (p. 3-6) (19:55)
Ways to Represent a Function (p. 7-8) (15:38)
Finding the Domain of a Function (p. 9) (4:31)
Piecewise Functions (p. 10) (9:29)
Even vs Odd vs Neither (p. 11-12) (6:02)
Increasing, Decreasing, Constant (p. 13) (2:57)
Essential Functions (p. 14-19) (15:15)
Transforming Functions (p. 20-22) (12:22)
Combinations of Functions (p. 23) (4:11)
Composite Functions (p. 24) (12:52)
Exponential Functions (p. 25) (5:01)
Inverse Functions (p. 26-28) (11:28)
Logarithms (p. 29-31) (9:58)
Exponents and Logs - Side by Side (p. 32-36) (11:19)
Linear Functions (p. 37-39) (10:20)
Solving Inequalities (p. 40-41) (10:50)
Rational Functions (p. 42-45) (16:34)

Vector Fields Practice Exam Solutions

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