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The speaker verbally says "greatest common denominator," but the mathematical notation on the board is "gcd," which conventionally stands for "greatest common divisor." The context of dividing integers to find a common factor confirms that the intended concept is the greatest common divisor, and the spoken word is a verbal slip.

Conditions: The video discusses finding the common factor of two integers.; The board displays the notation gcd(a;b).; The procedure involves repeated integer division.

The accumulation idea applies to motion by integrating velocity over time. Integrating signed velocity gives net displacement, while integrating the absolute value of velocity gives total distance traveled.

Conditions: The variable of integration is time.; The integrand is velocity (signed or absolute).

The harmonic mean is used because time spent traveling varies inversely with speed for fixed distances. When driving at different speeds over equal legs, you spend significantly more time at the slower speed, weighting the overall average towards the lower value.

Conditions: Averaging rates (like speed); Fixed distance segments (equal legs)

The presenter factors out the common coefficient $\frac{2}{3}$ to make the subsequent arithmetic simplification easier. After evaluating the antiderivative at the upper and lower bounds, both terms inside the brackets contain the factor $\frac{2}{3}$.

Conditions: The expression to evaluate is $\frac{4}{9}\left[\frac{2}{3}\cdot 9^{3/2}-\frac{2}{3}\cdot 1^{3/2}\right]$.; Both bracketed terms contain the same factor $\frac{2}{3}$.

In the Euclidean algorithm, when the division process yields a remainder of zero, the greatest common divisor of the original two integers is the last non-zero remainder obtained. The algorithm stops at this point because the method is over, and the last non-zero remainder is guaranteed to divide both original numbers evenly.

Conditions: The Euclidean algorithm is applied to two integers.; The process of repeated long division is followed until a remainder of zero is reached.

A Maclaurin series is a special case of a Taylor series where the center of expansion is exactly 0. A Taylor series generalizes this by allowing the expansion to be centered at an arbitrary point a.

Conditions: The Taylor formula is taken with center a.; Setting $a=0$ recovers the displayed Maclaurin formula.

Tracking the images of the standard basis vectors determines the destination of every other point because any arbitrary vector can be expressed as a linear combination of these basis vectors. Since a linear transformation preserves vector addition and scalar multiplication, the transformation of the arbitrary vector is exactly the same linear combination applied to the transformed basis vectors.

Conditions: The transformation is linear.; Working in a 2D plane with standard basis vectors $\hat{i} = [1, 0]^T$ and $\hat{j} = [0, 1]^T$.

The directional derivative represents the rate of change of the function's output when the input is nudged infinitesimally in a chosen direction. Geometrically, it is the slope of the tangent line to the curve formed by intersecting the surface with a vertical plane passing through the point and parallel to the direction vector.

Conditions: The function is a scalar-valued function of two variables.; The direction is specified by a vector in the input plane.; The step size approaches zero.

Multiplying by a negative number flips the vector's direction while adjusting its length based on the absolute value of the scalar.

Conditions: Scalar $c < 0$; Non-zero initial vector

The formula P(A and B) = $P(A)P(B)$ is crossed out because it is only valid for independent events. The video emphasizes the general multiplication rule, P(A and B) = $P(A)P$(B|A), which works for both independent and dependent events.

Conditions: The events A and B may be dependent.; The general multiplication rule P(A and B) = $P(A)P$(B|A) applies regardless of independence (assuming $P(A) > 0$).

To compute the matrix for the composition $A \circ B$, apply the transformation $A$ to each column of the matrix $B$. The resulting vectors form the columns of the matrix $A \circ B$.

Conditions: A and B are both 3x3 matrices.; The operation is performed over the real numbers.; These are real linear maps with compatible input and output spaces, represented using standard column-vector coordinates.

The solution to the normal equations is considered the least-squares solution because it satisfies the necessary and sufficient condition for minimizing the residual norm $\|\vec{b} - A\vec{x}\|$. The derivation shows that minimizing this norm is equivalent to requiring the residual $A\vec{x}^* - \vec{b}$ to be orthogonal to the column space $C(A)$.

Conditions: The original system $A\vec{x}=\vec{b}$ may be inconsistent; $A^T A \vec{x}^* = A^T \vec{b}$ has a solution