Resit Prep

CalcLin Exam Essentials

This note contains the methods, formulas, decision rules, and common mistakes for the topics that must be mastered for the exam.

1. Stokes’ Theorem

Purpose

Stokes’ theorem converts a line integral around a closed curve into a surface integral of the curl:

\oint_{C} F \cdot d r = \iint_{S} (\nabla \times F) \cdot n d S

Here:

$C$ is the closed boundary curve of the surface $S$ .
$n$ is the oriented normal vector.
The orientation of $C$ must agree with $n$ according to the right-hand rule.

When to Use Stokes

Look for:

A closed line integral $\oint_{C} F \cdot d r$ .
A curve that is the boundary of a surface.
A surface integral involving $\nabla \times F$ .
A complicated curve but an easy surface spanning the curve.

Work Plan

Compute the curl:

\nabla \times F = | \begin{matrix} i & j & k \\ \partial_{x} & \partial_{y} & \partial_{z} \\ F_{1} & F_{2} & F_{3} \end{matrix} |

Thus,

\nabla \times F = [\begin{matrix} \frac{\partial F_{3}}{\partial y} - \frac{\partial F_{2}}{\partial z} \\ \frac{\partial F_{1}}{\partial z} - \frac{\partial F_{3}}{\partial x} \\ \frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y} \end{matrix}]

Choose the easiest surface $S$ with boundary $C$ .
Find the correctly oriented normal vector.
Evaluate:

\iint_{S} (\nabla \times F) \cdot n d S

Surface Written as $z = g (x, y)$

For upward orientation:

n d S = [\begin{matrix} - g_{x} \\ - g_{y} \\ 1 \end{matrix}] d x d y

For downward orientation, multiply by $- 1$ .

Orientation

Curl the fingers of your right hand in the direction of $C$ :

Your thumb gives the positive normal direction.
Viewed from the tip of $n$ , the positive curve orientation is counterclockwise.
Reversing the curve direction changes the sign.

Important Shortcut

You may replace the original surface with any easier surface having the same boundary, as long as the orientation is preserved.

Common Mistakes

Using Stokes for a curve that is not closed.
Computing divergence instead of curl.
Forgetting the orientation.
Using a normal vector with the wrong sign.
Integrating over the boundary instead of the interior surface.

2. Gauss’ Divergence Theorem

Purpose

The divergence theorem converts the outward flux through a closed surface into a triple integral over the enclosed volume:

\iint_{\partial V} F \cdot n d S = ∭_{V} \nabla \cdot F d V

The surface $\partial V$ must be closed and $n$ must point outward.

When to Use Gauss

Look for:

Flux through a closed surface.
Phrases such as “outward flux.”
A sphere, cylinder, box, or another enclosed solid.
A direct surface integral that would require several separate pieces.

Divergence Formula

For

F = [\begin{matrix} F_{1} \\ F_{2} \\ F_{3} \end{matrix}]

the divergence is

\nabla \cdot F = \frac{\partial F_{1}}{\partial x} + \frac{\partial F_{2}}{\partial y} + \frac{\partial F_{3}}{\partial z}

Work Plan

Check that the surface is closed.
Compute $\nabla \cdot F$ .
Describe the enclosed volume $V$ .
Choose suitable coordinates.
Evaluate:

∭_{V} \nabla \cdot F d V

Open Surfaces

Gauss cannot be applied directly to an open surface. Instead:

Add a cap to close the surface.
Apply Gauss to the entire closed surface.
Subtract the flux through the added cap.

Flux through original surface = total closed flux - cap flux

Common Coordinate Systems

Cylindrical Coordinates

x = r \cos θ, y = r \sin θ, z = z

d V = r d r d θ d z

Spherical Coordinates

Using $ϕ$ as the angle from the positive $z$ -axis:

x = ρ \sin ϕ \cos θ

y = ρ \sin ϕ \sin θ

z = ρ \cos ϕ

d V = ρ^{2} \sin ϕ d ρ d ϕ d θ

Common Mistakes

Applying Gauss to an open surface.
Forgetting that the normal must point outward.
Forgetting the Jacobian.
Computing curl instead of divergence.
Forgetting to subtract an added cap.

3. Stokes or Gauss?

Use Stokes When

You have circulation around a closed curve.
The integral is $\oint_{C} F \cdot d r$ .
You need $\nabla \times F$ .
A curve bounds a surface.

Use Gauss When

You have outward flux through a closed surface.
The integral is $\iint_{\partial V} F \cdot n d S$ .
You need $\nabla \cdot F$ .
A surface encloses a volume.

Memory Rule

Stokes: boundary curve ⟷ surface

Gauss: closed surface ⟷ volume

4. Direct Flux Integrals

Definition

The flux of $F$ through an oriented surface $S$ is

\iint_{S} F \cdot n d S

Flux measures how much of the vector field passes through the surface.

Parametrized Surface

Suppose

r (u, v) = [\begin{matrix} x (u, v) \\ y (u, v) \\ z (u, v) \end{matrix}]

Then

r_{u} = \frac{\partial r}{\partial u}, r_{v} = \frac{\partial r}{\partial v}

The oriented surface element is

n d S = (r_{u} \times r_{v}) d u d v

Therefore,

\iint_{S} F \cdot n d S = \iint_{D} F (r (u, v)) \cdot (r_{u} \times r_{v}) d u d v

Work Plan

Parametrize the surface.
Compute $r_{u}$ and $r_{v}$ .
Compute $r_{u} \times r_{v}$ .
Check the orientation.
Substitute the parametrization into $F$ .
Take the dot product.
Integrate over the parameter domain.

Graph $z = g (x, y)$

Use

r (x, y) = [\begin{matrix} x \\ y \\ g (x, y) \end{matrix}]

Then

r_{x} \times r_{y} = [\begin{matrix} - g_{x} \\ - g_{y} \\ 1 \end{matrix}]

Thus, for upward orientation,

\iint_{S} F \cdot n d S = \iint_{D} F (x, y, g (x, y)) \cdot [\begin{matrix} - g_{x} \\ - g_{y} \\ 1 \end{matrix}] d x d y

Orientation Check

If the normal points in the wrong direction, multiply the final normal vector by $- 1$ .

For example:

Upward means the $z$ -component must be positive.
Downward means the $z$ -component must be negative.
Outward means the normal points away from the enclosed region.

Scalar Surface Integral Versus Flux

Scalar surface integral:

\iint_{S} f d S = \iint_{D} f (r (u, v)) ‖ r_{u} \times r_{v} ‖ d u d v

Flux integral:

\iint_{S} F \cdot n d S = \iint_{D} F (r (u, v)) \cdot (r_{u} \times r_{v}) d u d v

For flux, do not take the magnitude of the cross product.

5. Subspaces

Definition

A subset $H$ of a vector space is a subspace when it satisfies all three properties:

The zero vector is in $H$ .
$H$ is closed under addition.
$H$ is closed under scalar multiplication.

Subspace Test

Property 1: Zero Vector

Check:

0 \in H

If the zero vector is not in the set, the set is immediately not a subspace.

Property 2: Closure Under Addition

Take arbitrary $u, v \in H$ and show:

u + v \in H

Property 3: Closure Under Scalar Multiplication

Take arbitrary $u \in H$ and $c \in R$ and show:

c u \in H

Fast Recognition

The solution set of a homogeneous linear system

A x = 0

is always a subspace.

The solution set of

A x = b, b \neq 0

is generally not a subspace because it usually does not contain the zero vector.

Common Non-Subspace Conditions

Sets involving the following are suspicious:

Constants not equal to zero.
Absolute values.
Products such as $x y = 0$ .
Inequalities.
Restrictions such as $x \geq 0$ .
Nonlinear equations.

Disproving a Subspace

You only need one failed property. Useful methods:

Show $0 \notin H$ .
Find $u, v \in H$ for which $u + v \notin H$ .
Find $u \in H$ and $c \in R$ for which $c u \notin H$ .

6. Linear Dependence and Independence

Definition

Vectors $v_{1}, \dots, v_{p}$ are linearly independent when

c_{1} v_{1} + \dots + c_{p} v_{p} = 0

has only the trivial solution

c_{1} = \dots = c_{p} = 0

If a nontrivial solution exists, the vectors are linearly dependent.

Matrix Method

Place the vectors as columns:

A = [\begin{matrix} v_{1} & v_{2} & \dots & v_{p} \end{matrix}]

Solve

A x = 0

Conclusion

Pivot in every column $\Rightarrow$ linearly independent.
At least one free variable $\Rightarrow$ linearly dependent.

Immediate Dependence Rules

A set is automatically linearly dependent if:

It contains the zero vector.
One vector is a multiple of another.
One vector is a linear combination of the others.
There are more than $n$ vectors in $R^{n}$ .

Square Matrix Shortcut

For $n$ vectors in $R^{n}$ :

det (A) \neq 0 ⟺ vectors are linearly independent

det (A) = 0 ⟺ vectors are linearly dependent

Spanning Versus Independence

For vectors in $R^{m}$ :

Pivot in every column $\Rightarrow$ independent.
Pivot in every row $\Rightarrow$ span $R^{m}$ .

For exactly $n$ vectors in $R^{n}$ , the following are equivalent:

They are linearly independent.
They span $R^{n}$ .
They form a basis.
The matrix is invertible.
The determinant is nonzero.
There is a pivot in every row and column.

7. Rank Theorem

For an $m \times n$ matrix $A$ :

rank (A) + nullity (A) = n

Here:

$rank (A)$ is the number of pivots.
$nullity (A) = \dim (Nul A)$ .
$n$ is the number of columns.

Therefore,

number of pivot variables + number of free variables = number of variables

Important Dimensions

\dim (Col A) = rank (A)

\dim (Row A) = rank (A)

\dim (Nul A) = n - rank (A)

Basis for Column Space

Row reduce $A$ .
Identify the pivot column numbers.
Take those columns from the original matrix.

Never take the column-space basis from the row-reduced matrix.

Basis for Null Space

Row reduce $A$ .
Identify the free variables.
Express the pivot variables using the free variables.
Write the solution in parametric vector form.
The vectors multiplying the free parameters form a basis for $Nul A$ .

Example Structure

Suppose

x = s v_{1} + t v_{2}

Then

Nul A = Span {v_{1}, v_{2}}

and

nullity (A) = 2

8. Projections

Projection onto One Vector

The projection of $y$ onto a nonzero vector $u$ is

{proj}_{u} y = \frac{y \cdot u}{u \cdot u} u

If $u$ is a unit vector:

{proj}_{u} y = (y \cdot u) u

Orthogonal Decomposition

A vector $y$ can be written as

y = \hat{y} + z

where:

$\hat{y}$ is the projection onto the subspace.
$z$ is perpendicular to the subspace.

For projection onto $u$ :

\hat{y} = {proj}_{u} y

z = y - \hat{y}

The error vector satisfies

z \cdot u = 0

Projection onto an Orthogonal Basis

If ${u_{1}, \dots, u_{p}}$ is an orthogonal basis for $W$ , then

{proj}_{W} y = \sum_{i = 1}^{p} \frac{y \cdot u_{i}}{u_{i} \cdot u_{i}} u_{i}

Projection onto an Orthonormal Basis

If ${q_{1}, \dots, q_{p}}$ is orthonormal, then

{proj}_{W} y = \sum_{i = 1}^{p} (y \cdot q_{i}) q_{i}

Projection Matrix

If the columns of $Q$ are orthonormal, then

P = Q Q^{T}

and

\hat{y} = Q Q^{T} y

For a general full-column-rank matrix $A$ :

P = A (A^{T} A)^{- 1} A^{T}

9. Gram–Schmidt

Purpose

Gram–Schmidt converts a linearly independent set

{a_{1}, a_{2}, \dots, a_{p}}

into an orthogonal or orthonormal basis for the same subspace.

Orthogonal Version

Start with

v_{1} = a_{1}

Then

v_{2} = a_{2} - \frac{a_{2} \cdot v_{1}}{v_{1} \cdot v_{1}} v_{1}

For the third vector:

v_{3} = a_{3} - \frac{a_{3} \cdot v_{1}}{v_{1} \cdot v_{1}} v_{1} - \frac{a_{3} \cdot v_{2}}{v_{2} \cdot v_{2}} v_{2}

In general:

v_{k} = a_{k} - \sum_{j = 1}^{k - 1} {proj}_{v_{j}} a_{k}

Normalize

After finding the orthogonal vectors:

q_{i} = \frac{v_{i}}{∥ v_{i} ∥}

The vectors $q_{i}$ form an orthonormal basis.

Work Plan

Set $v_{1} = a_{1}$ .
Remove from $a_{2}$ its projection onto $v_{1}$ .
Remove from $a_{3}$ its projections onto both $v_{1}$ and $v_{2}$ .
Continue similarly.
Normalize only at the end unless unit vectors are specifically needed earlier.

Check

For an orthogonal basis:

v_{i} \cdot v_{j} = 0 (i \neq j)

For an orthonormal basis:

q_{i} \cdot q_{j} = {\begin{cases} 1, & i = j \\ 0, & i \neq j \end{cases}

Common Mistakes

Projecting onto the original vectors instead of the new orthogonal vectors.
Forgetting to subtract all previous projections.
Using $∥ v ∥$ instead of $∥ v ∥^{2}$ in the projection denominator.
Normalizing before completing the subtraction incorrectly.
Continuing when the original vectors are linearly dependent and obtaining a zero vector.

10. Least Squares

Purpose

The system

A x = b

may have no exact solution when $b \notin Col A$ .

The least-squares solution $\hat{x}$ makes

∥ A x - b ∥

as small as possible.

Geometric Interpretation

The vector

A \hat{x}

is the orthogonal projection of $b$ onto $Col A$ .

The residual is

r = b - A \hat{x}

The residual is orthogonal to every column of $A$ :

A^{T} r = 0

Normal Equations

Starting from

A^{T} (b - A \hat{x}) = 0

we obtain

A^{T} A \hat{x} = A^{T} b

Work Plan

Compute $A^{T} A$ .
Compute $A^{T} b$ .
Solve:

A^{T} A \hat{x} = A^{T} b

Compute the fitted vector:

\hat{b} = A \hat{x}

Compute the residual:

r = b - \hat{b}

Check:

A^{T} r = 0

Explicit Formula

If the columns of $A$ are linearly independent:

\hat{x} = (A^{T} A)^{- 1} A^{T} b

Do not use this formula if $A^{T} A$ is not invertible.

Least-Squares Line

For a line

y = β_{0} + β_{1} x

the design matrix is

A = [\begin{matrix} 1 & x_{1} \\ 1 & x_{2} \\ ⋮ & ⋮ \\ 1 & x_{m} \end{matrix}], x = [\begin{matrix} β_{0} \\ β_{1} \end{matrix}], b = [\begin{matrix} y_{1} \\ y_{2} \\ ⋮ \\ y_{m} \end{matrix}]

Solve the normal equations for $β_{0}$ and $β_{1}$ .

Common Mistakes

Solving $A x = b$ directly when it is inconsistent.
Writing $A A^{T}$ instead of $A^{T} A$ .
Forgetting that the residual is $b - A \hat{x}$ .
Assuming the residual itself is zero.
Forgetting that $A^{T} r = 0$ .

11. Sequence and Series Convergence

Sequence Convergence

A sequence ${a_{n}}$ converges to $L$ if

lim_{n \to \infty} a_{n} = L

A necessary condition for a series

\sum_{n = 1}^{\infty} a_{n}

to converge is

lim_{n \to \infty} a_{n} = 0

lim_{n \to \infty} a_{n} \neq 0

the series diverges.

However,

lim_{n \to \infty} a_{n} = 0

does not prove convergence.

Example:

\sum_{n = 1}^{\infty} \frac{1}{n}

diverges even though $1 / n \to 0$ .

12. Standard Series

Geometric Series

\sum_{n = 0}^{\infty} a r^{n}

converges when

| r | < 1

Its sum is

\sum_{n = 0}^{\infty} a r^{n} = \frac{a}{1 - r}

It diverges when

| r | \geq 1

$p$ -Series

\sum_{n = 1}^{\infty} \frac{1}{n^{p}}

converges when

p > 1

and diverges when

p \leq 1

Harmonic Series

\sum_{n = 1}^{\infty} \frac{1}{n}

diverges.

13. Choosing a Convergence Test

Step 1: Term Test

Always begin by checking:

lim_{n \to \infty} a_{n}

If the limit is not zero, the series diverges immediately.

Step 2: Recognize the Form

Geometric form $\Rightarrow$ geometric-series test.
$1 / n^{p}$ form $\Rightarrow$ $p$ -series test.
Positive decreasing function involving powers or logarithms $\Rightarrow$ integral test.
Alternating signs $(- 1)^{n}$ or $(- 1)^{n + 1}$ $\Rightarrow$ alternating-series test.
Factorials or exponentials $\Rightarrow$ ratio test.
Terms raised to the power $n$ $\Rightarrow$ root test.
Similarity to a familiar positive series $\Rightarrow$ comparison or limit comparison.

14. Integral Test

Conditions

Suppose

a_{n} = f (n)

where $f$ is:

Positive.
Continuous.
Decreasing.

Then

\sum_{n = N}^{\infty} a_{n} and \int_{N}^{\infty} f (x) d x either both converge or both diverge.

Work Plan

Identify $f (x)$ such that $f (n) = a_{n}$ .
State that $f$ is positive, continuous, and decreasing.
Compute:

\int_{N}^{\infty} f (x) d x = lim_{b \to \infty} \int_{N}^{b} f (x) d x

Conclude:
- Finite improper integral $\Rightarrow$ series converges.
- Infinite improper integral $\Rightarrow$ series diverges.

Example

For

\sum_{n = 1}^{\infty} \frac{1}{n^{2}}

use

f (x) = \frac{1}{x^{2}}

Then

\int_{1}^{\infty} \frac{1}{x^{2}} d x = {[- \frac{1}{x}]}_{1}^{\infty} = 1

Therefore, the series converges.

Important

The integral test normally determines convergence, not the exact value of the series.

15. Integral-Test Error Bounds

Let

S = \sum_{n = N}^{\infty} a_{n}

and let

S_{n} = \sum_{k = N}^{n} a_{k}

The remainder is

R_{n} = S - S_{n}

If the integral-test conditions hold, then

\int_{n + 1}^{\infty} f (x) d x \leq R_{n} \leq \int_{n}^{\infty} f (x) d x

Maximum Error

The upper bound gives the maximum possible error:

| R_{n} | \leq \int_{n}^{\infty} f (x) d x

Better Estimate for the Sum

Since

S = S_{n} + R_{n}

and

L_{n} \leq R_{n} \leq U_{n}

we obtain

S_{n} + L_{n} \leq S \leq S_{n} + U_{n}

A better estimate is the midpoint:

S \approx S_{n} + \frac{L_{n} + U_{n}}{2}

The maximum error of this midpoint estimate is

maximum error = \frac{U_{n} - L_{n}}{2}

Here,

L_{n} = \int_{n + 1}^{\infty} f (x) d x

U_{n} = \int_{n}^{\infty} f (x) d x

Finding the Required Number of Terms

To guarantee an error smaller than $ε$ , solve

\int_{n}^{\infty} f (x) d x < ε

Then round $n$ upward to the next integer.

16. Alternating-Series Test

An alternating series has the form

\sum_{n = 1}^{\infty} (- 1)^{n} b_{n} or \sum_{n = 1}^{\infty} (- 1)^{n + 1} b_{n}

where $b_{n} \geq 0$ .

The series converges if:

$b_{n + 1} \leq b_{n}$ eventually.
$lim_{n \to \infty} b_{n} = 0$ .

Alternating-Series Error Bound

If the alternating-series test applies, then

| R_{n} | = | S - S_{n} | \leq b_{n + 1}

The error is at most the magnitude of the first omitted term.

Finding the Number of Terms

To guarantee

| R_{n} | < ε

solve

b_{n + 1} < ε

Sign of the Error

The exact sum lies between consecutive partial sums:

S_{n} and S_{n + 1}

If the first omitted term is positive, the exact answer lies above $S_{n}$ .

If the first omitted term is negative, the exact answer lies below $S_{n}$ .

17. Absolute and Conditional Convergence

A series

\sum a_{n}

converges absolutely if

\sum | a_{n} |

converges.

A series converges conditionally if

\sum a_{n}

converges but

\sum | a_{n} |

diverges.

Procedure for Alternating Series

Test

\sum | a_{n} |

If it converges, the original series converges absolutely.
If it diverges, test the original alternating series.
If the original converges, it converges conditionally.
If the original also diverges, the series diverges.

Important Result

Absolute convergence implies convergence.

18. Ratio Test

For

\sum a_{n}

compute

L = lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} |

Then:

L < 1 \Rightarrow absolute convergence

L > 1 \Rightarrow divergence

L = 1 \Rightarrow inconclusive

The ratio test is especially useful for:

Factorials.
Exponentials.
Products.
Power series.

19. Power Series

A power series centred at $a$ has the form

\sum_{n = 0}^{\infty} c_{n} (x - a)^{n}

It normally has:

Convergence for $| x - a | < R$ .
Divergence for $| x - a | > R$ .
Endpoints that must be tested separately.

Work Plan

Apply the ratio test or root test.
Solve the resulting inequality for $| x - a |$ .
Determine the radius $R$ .
Write the open interval.
Test each endpoint separately.
State the full interval of convergence.

Endpoint Warning

The ratio test usually gives no conclusion at the endpoints. Each endpoint may:

Converge absolutely.
Converge conditionally.
Diverge.

20. Taylor and Maclaurin Series

Taylor Series

The Taylor series of $f$ about $x = a$ is

f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (a)}{n!} (x - a)^{n}

Maclaurin Series

For $a = 0$ :

f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} x^{n}

Important Standard Series

Geometric

\frac{1}{1 - x} = \sum_{n = 0}^{\infty} x^{n}, | x | < 1

Exponential

e^{x} = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!}

Sine

\sin x = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{2 n + 1}}{(2 n + 1)!}

Cosine

\cos x = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{2 n}}{(2 n)!}

Natural Logarithm

\ln (1 + x) = \sum_{n = 1}^{\infty} (- 1)^{n + 1} \frac{x^{n}}{n}, - 1 < x \leq 1

Arctangent

\arctan x = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{2 n + 1}}{2 n + 1}, | x | \leq 1

Transforming Known Series

Starting from

\frac{1}{1 - x} = \sum_{n = 0}^{\infty} x^{n}

you may:

Replace $x$ by another expression.
Multiply the entire series.
Differentiate term by term.
Integrate term by term.

Always update the interval of convergence.

21. Taylor Error Bound

The degree- $n$ Taylor polynomial is

T_{n} (x) = \sum_{k = 0}^{n} \frac{f^{(k)} (a)}{k!} (x - a)^{k}

The remainder is

R_{n} (x) = f (x) - T_{n} (x)

Lagrange Error Bound

| f^{(n + 1)} (t) | \leq M

for all $t$ between $a$ and $x$ , then

| R_{n} (x) | \leq \frac{M | x - a |^{n + 1}}{(n + 1)!}

Work Plan

Identify the degree $n$ .
Find $f^{(n + 1)}$ .
Find a maximum value $M$ on the interval between $a$ and $x$ .
Substitute into:

| R_{n} (x) | \leq \frac{M | x - a |^{n + 1}}{(n + 1)!}

Choosing the Degree

To guarantee an error below $ε$ , find the smallest $n$ for which

\frac{M | x - a |^{n + 1}}{(n + 1)!} < ε

22. Error-Bound Summary

Alternating Series

| R_{n} | \leq b_{n + 1}

Use the first omitted term.

Integral Test

\int_{n + 1}^{\infty} f (x) d x \leq R_{n} \leq \int_{n}^{\infty} f (x) d x

Maximum error using $S_{n}$ :

| R_{n} | \leq \int_{n}^{\infty} f (x) d x

Midpoint estimate:

S \approx S_{n} + \frac{1}{2} (\int_{n + 1}^{\infty} f (x) d x + \int_{n}^{\infty} f (x) d x) $ T e m p l a t e

Midpoint maximum error:

\frac{1}{2} (\int_{n}^{\infty} f (x) d x - \int_{n + 1}^{\infty} f (x) d x)

Taylor Polynomial

| R_{n} (x) | \leq \frac{M | x - a |^{n + 1}}{(n + 1)!}

23. Final Exam Decision Guide

Closed Line Integral

\oint_{C} F \cdot d r

Use Stokes’ theorem. Compute:

\nabla \times F

Outward Flux Through a Closed Surface

\iint_{\partial V} F \cdot n d S

Use Gauss’ theorem. Compute:

\nabla \cdot F

Flux Through an Open Surface

Parametrize the surface and use

\iint_{D} F (r (u, v)) \cdot (r_{u} \times r_{v}) d u d v

Alternatively, close the surface and use Gauss, then subtract the added part.

Is a Set a Subspace?

Check:

Zero vector.
Closure under addition.
Closure under scalar multiplication.

A homogeneous solution set is automatically a subspace.

Are Vectors Linearly Independent?

Put the vectors in the columns of a matrix and row reduce.

Pivot in every column $\Rightarrow$ independent.
Free variable $\Rightarrow$ dependent.

Need Rank or Nullity?

Use

rank (A) + nullity (A) = n

Need an Orthogonal Basis?

Use Gram–Schmidt:

v_{k} = a_{k} - \sum_{j = 1}^{k - 1} {proj}_{v_{j}} a_{k}

Inconsistent System but Need Best Approximation?

Use least squares:

A^{T} A \hat{x} = A^{T} b

Series with Alternating Signs?

Check:

$b_{n}$ decreases.
$b_{n} \to 0$ .

Error:

| R_{n} | \leq b_{n + 1}

Use the integral test.

Error:

\int_{n + 1}^{\infty} f (x) d x \leq R_{n} \leq \int_{n}^{\infty} f (x) d x

Factorials or Exponentials?

Use the ratio test.

Taylor Approximation Error?

Use

| R_{n} (x) | \leq \frac{M | x - a |^{n + 1}}{(n + 1)!}

24. Formula Sheet

Vector Calculus

\nabla \cdot F = F_{1 x} + F_{2 y} + F_{3 z}

\nabla \times F = [\begin{matrix} F_{3 y} - F_{2 z} \\ F_{1 z} - F_{3 x} \\ F_{2 x} - F_{1 y} \end{matrix}]

\oint_{C} F \cdot d r = \iint_{S} (\nabla \times F) \cdot n d S

\iint_{\partial V} F \cdot n d S = ∭_{V} \nabla \cdot F d V

n d S = (r_{u} \times r_{v}) d u d v

Linear Algebra

{proj}_{u} y = \frac{y \cdot u}{u \cdot u} u

rank (A) + nullity (A) = n

A^{T} A \hat{x} = A^{T} b

\hat{x} = (A^{T} A)^{- 1} A^{T} b

r = b - A \hat{x}

A^{T} r = 0

Series

\sum_{n = 0}^{\infty} a r^{n} = \frac{a}{1 - r}, | r | < 1

\sum_{n = 1}^{\infty} \frac{1}{n^{p}} converges iff p > 1

| R_{n} |_{alternating} \leq b_{n + 1}

\int_{n + 1}^{\infty} f (x) d x \leq R_{n} \leq \int_{n}^{\infty} f (x) d x

| R_{n} (x) |_{Taylor} \leq \frac{M | x - a |^{n + 1}}{(n + 1)!}

25. Last-Minute Checklist

Before the exam, make sure you can do the following without notes:

Recognize whether to use Stokes, Gauss, or a direct flux integral.
Compute curl and divergence correctly.
Parametrize a graph, cylinder, plane, and sphere.
Determine the correct orientation of a normal.
Test whether a set is a subspace.
Determine linear independence using pivots.
Find rank and nullity.
Find bases for column space and null space.
Project a vector onto a vector or subspace.
Carry out Gram–Schmidt for two or three vectors.
Form and solve the normal equations.
Compute and interpret a least-squares residual.
Recognize geometric and $p$ -series.
Apply the integral and alternating-series tests.
Distinguish absolute and conditional convergence.
Determine an interval of convergence and test endpoints.
Compute alternating, integral-test, and Taylor error bounds.
Determine how many terms are required for a stated maximum error.

CalcLin Exam Essentials

1. Stokes’ Theorem

Purpose

When to Use Stokes

Work Plan

Surface Written as z=g(x,y)

Orientation

Important Shortcut

Common Mistakes

2. Gauss’ Divergence Theorem

Purpose

When to Use Gauss

Divergence Formula

Work Plan

Open Surfaces

Common Coordinate Systems

Cylindrical Coordinates

Spherical Coordinates

Common Mistakes

3. Stokes or Gauss?

Use Stokes When

Use Gauss When

Memory Rule

4. Direct Flux Integrals

Definition

Parametrized Surface

Work Plan

Graph z=g(x,y)

Orientation Check

Scalar Surface Integral Versus Flux

5. Subspaces

Definition

Subspace Test

Property 1: Zero Vector

Property 2: Closure Under Addition

Property 3: Closure Under Scalar Multiplication

Fast Recognition

Common Non-Subspace Conditions

Disproving a Subspace

6. Linear Dependence and Independence

Definition

Matrix Method

Conclusion

Immediate Dependence Rules

Square Matrix Shortcut

Spanning Versus Independence

7. Rank Theorem

Important Dimensions

Basis for Column Space

Basis for Null Space

Example Structure

8. Projections

Projection onto One Vector

Orthogonal Decomposition

Projection onto an Orthogonal Basis

Projection onto an Orthonormal Basis

Projection Matrix

9. Gram–Schmidt

Purpose

Orthogonal Version

Normalize

Work Plan

Check

Common Mistakes

10. Least Squares

Purpose

Geometric Interpretation

Normal Equations

Work Plan

Explicit Formula

Least-Squares Line

Common Mistakes

11. Sequence and Series Convergence

Sequence Convergence

12. Standard Series

Geometric Series

p-Series

Harmonic Series

13. Choosing a Convergence Test

Step 1: Term Test

Surface Written as $z = g (x, y)$

Graph $z = g (x, y)$

$p$ -Series