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Define vectors in 3 space |
Perform vector operations |
Find equations for lines and planes in 3 space |
Describe and identify surfaces in space |
Convert to cylindrical and spherical coordinates |
Define a vector valued function |
Differentiate and integrate vector valued functions |
Find velocity and acceleration in 3 space |
Find the arc length and curvature of a space curve |
Find the tangent and normal vectors |
Define a function of several variables |
Find the Limit of a function of several variables |
Find partial derivatives of multivariable function |
Find the differential |
Use multivariable chain rule |
Find the directional derivative and the Gradient |
Find tangent plane and normal line |
Find extrema of multivariable function |
Find iterated integral and area in the plane |
Find double integral and volume |
Convert to polar and perform calculus operations |
Find surface area |
Set up and find triple integral |
Identify a matrix |
Perform elimination on a matrix to solve a system |
Define rules for matrix operations |
Define a linear combination. |
Find the inverse of a matrix to solve a system |
Perform LU factorization |
Describe the transpose and permutations of a matrix |
Define a vector space |
Define a vector subspace |
Find the Null space |
Find the complete solution |
Define span |
Define the rank of a matrix |
Determine independence basis and dimension of a matrix |
Determine orthogonality |
Perform projections |
Use determinate properties to solve a system |
Find permutations and cofactors |
Use Cramer's rule to solve a system |
Find Eigenvalues and Eigenvectors |
Form logical statements |
Determine truth values of logical statements |
Recognize quantified statements |
Develop a direct proof |
Develop an indirect proof |
Develop a proof by induction |
Understand and apply sequence formulas and notation |
Define Set Theory |
Identify properties of sets |
Use fundamental counting principals |
Create possibility trees |
Perform combinations and permutations |
Find basic discrete probabilities |
Find conditional probabilities |
Use Bayes' Formula |
Develop and use the Binomial Theorem |
Essential Questions
How are multivariable and single variable calculus similar? Different? |
What do vector operations produce? |
How is a vector valued function different than a regular function? |
What are tangent and normal vectors and how are they used? |
How is arc length and curvature found and what is it used for in calculus? |
What is a function in several variables? |
How are partial derivatives used to find the total differential? |
Why does the chain rule for multivariable calculus work? |
What is the directional derivative and gradient explain? |
What are the extrema for multivariable functions and how are they identified? |
What is an iterated integral? |
How does a double integral find volume? |
How are polar and rectangular coordinates related in 3 space? |
What is the structure of a triple integral and how are the limits of integration set up? |
What is a matrix? |
What is elimination and how is it used? |
What is a linear combination? |
What is LU decomposition and how is it used? |
What is the inverse of a matrix and how is it used? |
What is the transpose of a matrix? |
What are the common permutations of a matrix? |
What are the axioms that determine a vector space? |
What are the axioms that determine a vector subspace? |
What is the Null Space of a system? |
What is the Rank of a matrix? |
What is the Span of a matrix? |
What is the complete solution to Ax = b ? |
What is the determinant? |
What are cofactors and how are they used? |
What is Cramer's rule? |
How is a matrix used to find volume? |
What is an Eigenvalue? |
What is an Eigenvector? |
How is a matrix diagonalized? |
What is logic? |
What is a conditional statement? |
What is a valid or invalid argument? |
What is a quantifier? |
What is a direct proof? |
What is an indirect proof? |
What is a sequence? |
What is mathematical induction? |
How is strong induction used? |
How are set properties used to identify sets? |
What is the fundamental counting principle? |
How was the Binomial Theorem developed and how is it used? |
How are basic discrete probabilities calculated? |
Essential Vocabulary
Vector | Standard unit vector | Velocity | Acceleration |
Space coordinates | Dot product | Cross product | Orthogonal |
Projection | Triple scalar | Paraboloid | Cylindrical coordinates |
Spherical coordinates | Space curve | Vector valued function | Derivative |
Anti derivative | Tangent vector | Normal vector | Unit tangent vector |
Tangential component | Normal component | Circular helix | Arc length |
Curvature | Level curve | Contour map | Limit |
Continuity | Partial derivative | Total differential | Chain rule |
Directional derivative | Gradient | Tangent plane | Normal line |
Extrema | Saddle point | Second partial | Iterated integral |
Double integral | Polar coordinates | Surface area | Triple integral |
Matrix | Linear combination | Elimination | Block matrix |
Invertible | Singular | Nonsingular | Gauss-Jordan |
Lower triangular | Upper triangular | Diagonal | Transpose |
Permutation | Symmetric matrix | Vector space | Vector subspace |
Null space | Rank | Span | Complete solution |
Orthoganality | Determinant | Pivot | Cofactors |
Cramer's rule | Eigenvalue | Eigenvector | Dependent |
Independent | Logic | Conditional statement | Valid argument |
Invalid argument | Modus Tollens | Modus Ponens | Quantified statement |
Predicate | Direct proof | Indirect proof | Universal statement |
Bi-conditional | Sufficient conditions | Necessary conditions | Counterexample |
Existential statement | Contradiction | Sequence | Induction |
Strong induction | Set | Subset | Sample space |
Probability | Possibility tree | Disjoint sets | Repetition |
Combination | Permutation | Expected value | Independent event |
Units of Study
Vectors in space |
vector valued functions |
functions of several variables |
multiple integration |
Solving Linear systems |
vector spaces and subspaces |
determinants |
eigenvalues and eigenvectors |
Logic of compount statements |
logic of quantified statements |
elementary number theory |
methods of proof |
sequences and induction |
set theory |
counting and probability |
