Enrichment • Series
Multivariable Calculus (Calculus 3)
BL
SL
Samuel L
This series ended on December 17, 2021. All 1:1 and group chats related to this series are disabled 7 days after the last session.
Series Details
About
We will go through a multivariable Calculus class by working problems together in each class. The topics will cover parametric equations, polar coordinates, vectors in space, vector-valued functions, differentiation of functions of several variables, multiple integration, vector calculus, and second-order differential equations.
You should be knowledgeable in limits, derivatives, and integrals in the xy-plane before enrolling in this series. Having some knowledge in precalculus vectors can help but is not required.
The session times are set to two hours, estimating the time you might need to understand the entire topic. It might take less time than that. It will depend on how well is everyone getting the topic.
Tutor Qualifications
I took this class last fall in college and really enjoyed it. Hope you will enjoy it too!
✋ ATTENDANCE POLICY
You are free to attend any session you wish to. Attendance is not mandatory since the class is intended to compliment a calculus class you might be taking or have taken. Attendance to all classes is encouraged since (as all topics in math) they build upon each other.
Dates
September 3 - November 27
Learners
12 / 15
Total Sessions
22
About the Tutor
BL
SL
I am a Guatemalan living in the U.S. as an international student. I have always loved math and sciences, and I am always keen to help people who need help with them. I hope to be able to show you how beautiful math is.
View Samuel L's Profile
Upcoming Sessions
0
Past Sessions
22
3
Sep
O
Session 1
Orientation
This is a session to briefly introduce each other, especially those intending to stay for the entire class. After that, we will go over parametric equations and calculus of parametric curves.
4
Sep
EM
Session 2
Even More Math
We will go over polar coordinates, finding areas in polar coordinates, finding the arc length in polar coordinates, and conic sections.
10
Sep
EM
Session 3
Even More Math
PARAMETRIC EQUATIONS AND POLAR COORDINATES
Conic Sections
- Identify the equation of a parabola in standard form with given focus and directrix. - Identify the equation of an ellipse in standard form with given foci. - Identify the equation of a hyperbola in standard form with given foci. - Recognize a parabola, ellipse, or hyperbola from its eccentricity value. - Write the polar equation of a conic section with eccentricity e . - Identify when a general equation of degree two is a parabola, ellipse, or hyperbola.
11
Sep
EM
Session 4
Even More Math
VECTORS IN SPACE
Vectors in the Plane
- Describe a plane vector, using correct notation. - Perform basic vector operations (scalar multiplication, addition, subtraction). - Express a vector in component form. - Explain the formula for the magnitude of a vector. - Express a vector in terms of unit vectors. - Give two examples of vector quantities.
17
Sep
EM
Session 5
Even More Math
VECTORS IN SPACE
Vectors in Three Dimensions
- Describe three-dimensional space mathematically. - Locate points in space using coordinates. - Write the distance formula in three dimensions. - Write the equations for simple planes and spheres. - Perform vector operations in R3.
The Dot Product
- Calculate the dot product of two given vectors. - Determine whether two given vectors are perpendicular. - Find the direction cosines of a given vector. - Explain what is meant by the vector projection of one vector onto another vector, and describe how to compute it. - Calculate the work done by a given force.
18
Sep
EM
Session 6
Even More Math
VECTORS IN SPACE
The Cross Product
- Calculate the cross product of two given vectors. - Use determinants to calculate a cross product. - Find a vector orthogonal to two given vectors. - Determine areas and volumes by using the cross product. - Calculate the torque of a given force and position vector.
24
Sep
EM
Session 7
Even More Math
VECTORS IN SPACE
Equations of Lines and Planes in Space
- Write the vector, parametric, and symmetric equations of a line through a given point in a given direction, and a line through two given points. - Find the distance from a point to a given line. - Write the vector and scalar equations of a plane through a given point with a given normal. - Find the distance from a point to a given plane. - Find the angle between two planes.
Quadric Surfaces
- Identify a cylinder as a type of three-dimensional surface. - Recognize the main features of ellipsoids, paraboloids, and hyperboloids. - Use traces to draw the intersections of quadric surfaces with the coordinate planes.
25
Sep
EM
Session 8
Even More Math
VECTORS IN SPACE
Cylindrical and Spherical Coordinates
- Convert from cylindrical to rectangular coordinates. - Convert from rectangular to cylindrical coordinates. - Convert from spherical to rectangular coordinates. - Convert from rectangular to spherical coordinates.
1
Oct
EM
Session 9
Even More Math
VECTOR-VALUED FUNCTIONS
Vector-Valued Functions and Space Curves
- Write the general equation of a vector-valued function in component form and unit-vector form. - Recognize parametric equations for a space curve. - Describe the shape of a helix and write its equation. - Define the limit of a vector-valued function.
Calculus of Vector-Valued Functions
-Write an expression for the derivative of a vector-valued function. - Find the tangent vector at a point for a given position vector. - Find the unit tangent vector at a point for a given position vector and explain its significance. - Calculate the definite integral of a vector-valued function.
2
Oct
EM
Session 10
Even More Math
VECTOR-VALUED FUNCTIONS
Arc Length and Curvature
- Determine the length of a particle’s path in space by using the arc-length function. - Explain the meaning of the curvature of a curve in space and state its formula. - Describe the meaning of the normal and binormal vectors of a curve in space.
8
Oct
EM
Session 11
Even More Math
VECTOR-VALUED FUNCTIONS
Arc Length and Curvature
- Determine the length of a particle’s path in space by using the arc-length function. - Explain the meaning of the curvature of a curve in space and state its formula. - Describe the meaning of the normal and binormal vectors of a curve in space.
9
Oct
EM
Session 12
Even More Math
VECTOR-VALUED FUNCTIONS
Motion in Space
- Describe the velocity and acceleration vectors of a particle moving in space. - Explain the tangential and normal components of acceleration. - State Kepler’s laws of planetary motion.
15
Oct
EM
Session 13
Even More Math
DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES
Functions of Several Variables
- Recognize a function of two variables and identify its domain and range. - Sketch a graph of a function of two variables. - Sketch several traces or level curves of a function of two variables. - Recognize a function of three or more variables and identify its level surfaces.
Limits and Continuity
- Calculate the limit of a function of two variables. - Learn how a function of two variables can approach different values at a boundary point, depending on the path of approach. - State the conditions for continuity of a function of two variables. - Verify the continuity of a function of two variables at a point. - Calculate the limit of a function of three or more variables and verify the continuity of the function at a point.
16
Oct
EM
Session 14
Even More Math
DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES
Partial Derivatives
- Calculate the partial derivatives of a function of two variables. - Calculate the partial derivatives of a function of more than two variables. - Determine the higher-order derivatives of a function of two variables. - Explain the meaning of a partial differential equation and give an example.
29
Oct
EM
Session 15
Even More Math
DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES
Tangent Planes and Linear Approximations
- Determine the equation of a plane tangent to a given surface at a point. - Use the tangent plane to approximate a function of two variables at a point. - Explain when a function of two variables is differentiable. - Use the total differential to approximate the change in a function of two variables.
The Chain Rule
- State the chain rules for one or two independent variables. - Use tree diagrams as an aid to understanding the chain rule for several independent and intermediate variables. - Perform implicit differentiation of a function of two or more variables.
30
Oct
EM
Session 16
Even More Math
DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES
Directional Derivatives and the Gradient
- Determine the directional derivative in a given direction for a function of two variables. - Determine the gradient vector of a given real-valued function. - Explain the significance of the gradient vector with regard to direction of change along a surface. - Use the gradient to find the tangent to a level curve of a given function. - Calculate directional derivatives and gradients in three dimensions.
Maxima/Minima Problems
- Use partial derivatives to locate critical points for a function of two variables. - Apply a second derivative test to identify a critical point as a local maximum, local minimum, or saddle point for a function of two variables. - Examine critical points and boundary points to find absolute maximum and minimum values for a function of two variables.
5
Nov
EM
Session 17
Even More Math
DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES
Lagrange Multipliers
- Use the method of Lagrange multipliers to solve optimization problems with one constraint. - Use the method of Lagrange multipliers to solve optimization problems with two constraints.
6
Nov
EM
Session 18
Even More Math
MULTIPLE INTEGRATION
Double Integrals over Rectangular Regions
- Recognize when a function of two variables is integrable over a rectangular region. - Recognize and use some of the properties of double integrals. - Evaluate a double integral over a rectangular region by writing it as an iterated integral. - Use a double integral to calculate the area of a region, volume under a surface, or average value of a function over a plane region.
19
Nov
EM
Session 19
Even More Math
MULTIPLE INTEGRATION
Double Integrals over General Regions
- Recognize when a function of two variables is integrable over a general region. - Evaluate a double integral by computing an iterated integral over a region bounded by two vertical lines and two functions of x, or two horizontal lines and two functions of y. - Simplify the calculation of an iterated integral by changing the order of integration. - Use double integrals to calculate the volume of a region between two surfaces or the area of a plane region. - Solve problems involving double improper integrals.
20
Nov
EM
Session 20
Even More Math
MULTIPLE INTEGRATION
Double Integrals in Polar Coordinates
- Recognize the format of a double integral over a polar rectangular region. - Evaluate a double integral in polar coordinates by using an iterated integral. - Recognize the format of a double integral over a general polar region. - Use double integrals in polar coordinates to calculate areas and volumes.
Triple Integrals
- Recognize when a function of three variables is integrable over a rectangular box. - Evaluate a triple integral by expressing it as an iterated integral. - Recognize when a function of three variables is integrable over a closed and bounded region. - Simplify a calculation by changing the order of integration of a triple integral. - Calculate the average value of a function of three variables.
26
Nov
EM
Session 21
Even More Math
MULTIPLE INTEGRATION
Triple Integrals in Cylindrical and Spherical Coordinates
- Evaluate a triple integral by changing to cylindrical coordinates. - Evaluate a triple integral by changing to spherical coordinates.
27
Nov
EM
Session 22
Even More Math
MULTIPLE INTEGRATION
Calculating Centers of Mass and Moments of Inertia
- Use double integrals to locate the center of mass of a two-dimensional object. - Use double integrals to find the moment of inertia of a two-dimensional object. - Use triple integrals to locate the center of mass of a three-dimensional object.