LAC Calculus & Linear Algebra Flashcards

💡 Intuition: In real-world engineering, systems depend on multiple independent variables (e.g. Temperature $T(x,y,z)$, Cylinder volume $V(r,h)$). An ordinary derivative $\frac{dy}{dx}$ only works on a 2D line. A partial derivative measures instantaneous slope along a multi-dimensional surface.

📐 Core Rule: To compute $\frac{\partial z}{\partial x}$, treat $y$ as a strictly frozen constant. To compute $\frac{\partial z}{\partial y}$, treat $x$ as a strictly frozen constant.

🎯 Exam Cue: Whenever an expression has 2+ independent variables: $\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h}$.

💡 Intuition: A function is homogeneous if scaling every input by a scalar $t$ scales the entire output uniformly by $t^n$: $f(tx, ty) = t^n f(x, y)$.

📐 Core Rule: Factor out $x$ from numerator ($x^p$) and denominator ($x^q$). If all remaining internal terms combine exclusively into powers of $(y/x)$, degree is $n = p - q$.

🎯 Exam Cue: If an isolated constant or mismatched power remains (e.g. $\frac{x^2+y^3}{x+y}$), the function is non-homogeneous and Euler's theorem cannot be used.

💡 Intuition: Euler's theorem lets you evaluate the weighted sum of partial derivatives without doing any manual differentiation.

📐 Core Rule: $$x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n \cdot u$$ (For 3 variables: $x u_x + y u_y + z u_z = n u$).

🎯 Exam Cue: When an exam problem asks to evaluate $x u_x + y u_y$, never differentiate manually. Identify the degree $n$ and write $n \cdot u$.

💡 Intuition: Extends the Euler shortcut to second-order partial derivatives, avoiding lengthy product rule calculations.

📐 Core Rule: $$x^2 \frac{\partial^2 u}{\partial x^2} + 2xy \frac{\partial^2 u}{\partial x \partial y} + y^2 \frac{\partial^2 u}{\partial y^2} = n(n - 1) u$$

🎯 Exam Cue: Requires continuous second-order derivatives so Clairaut's mixed partial identity ($u_{xy} = u_{yx}$) holds.

💡 Intuition: $u$ itself is not homogeneous, but moving the non-linear outer function to the other side makes $z = \sin(u)$ homogeneous with degree $n = 1 - \frac{1}{2} = \frac{1}{2}$.

📐 Core Rule: $$x u_x + y u_y = n \frac{F(u)}{F'(u)} \equiv G(u)$$ For $F(u) = \sin u$, $G(u) = n \tan u$.

🎯 Exam Cue: Common cases: If $F(u) = \tan u \implies G(u) = \frac{n}{2}\sin(2u)$. If $F(u) = \ln u \implies G(u) = n$.

💡 Intuition: Evaluates second-order expressions for composite homogeneous functions using only the first-order result $G(u)$ without messy second derivatives.

📐 Core Rule: $$x^2 u_{xx} + 2xy u_{xy} + y^2 u_{yy} = G(u)\big[G'(u) - 1\big]$$

🎯 Exam Cue: If $G(u) = \tan u$, then $G'(u) = \sec^2 u$, giving: $\tan u(\sec^2 u - 1) = \tan^3 u$.

💡 Intuition: The order of differentiation ($x$ then $y$ versus $y$ then $x$) does not matter for smooth, well-behaved surfaces.

📐 Core Rule: $$\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial^2 f}{\partial x \partial y}$$ Condition: Both mixed partial derivatives must exist and be continuous at the point.

🎯 Exam Cue: Frequently asked in 2-mark definitions: "State Schwarz's / Clairaut's Theorem on mixed partials".

💡 Intuition: Represents the best linear approximation (tangent flat surface) to a curved 3D surface at a given point.

📐 Core Rule: Total Differential: $$dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy$$ Tangent Plane: $$z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$$

🎯 Exam Cue: Used for engineering error propagation: $\delta z \approx f_x \delta x + f_y \delta y$.

💡 Intuition: The local area or volume magnification factor when transforming coordinates from one coordinate system to another.

📐 Core Rule: $$J = \begin{vmatrix} u_x & u_y \\ v_x & v_y \end{vmatrix} = u_x v_y - u_y v_x$$ For polar $x=r\cos\theta, y=r\sin\theta$, $J = r \implies dx\,dy = r\,dr\,d\theta$.

🎯 Exam Cue: In multiple integration, coordinate changes always multiply the differential element by $|J|$.

💡 Intuition: Tells you if coordinate systems can be inverted, or if two equations are secretly redundant.

📐 Core Rule: Reciprocal Property: $$\frac{\partial(u, v)}{\partial(x, y)} \cdot \frac{\partial(x, y)}{\partial(u, v)} = 1 \quad (J \cdot J' = 1)$$ Functional Dependence: $$\frac{\partial(u, v)}{\partial(x, y)} = 0 \iff \phi(u, v) = 0$$

🎯 Exam Cue: If asked "Prove $u$ and $v$ are functionally related", show $J = 0$, then eliminate $x,y$ to write the explicit relation.

💡 Intuition: Approximates any multivariable surface near a known point using polynomials of increasing degree.

📐 Core Rule: $$f(x+h, y+k) = f(x,y) + [h f_x + k f_y] + \frac{1}{2!}[h^2 f_{xx} + 2hk f_{xy} + k^2 f_{yy}] + \dots$$

🎯 Exam Cue: Maclaurin series is the special case centered at the origin $(0, 0)$.

💡 Intuition: Evaluates 3D surface curvature along all directions to find peaks, valleys, or saddle passes.

📐 Core Rule: Calculate discriminant $\Delta = rt - s^2$: 1. $\Delta > 0$ and $r < 0 \implies$ Local Maximum (Peak) 2. $\Delta > 0$ and $r > 0 \implies$ Local Minimum (Valley) 3. $\Delta < 0 \implies$ Saddle Point (Curves up one way, down another) 4. $\Delta = 0 \implies$ Inconclusive (Boundary analysis needed)

🎯 Exam Cue: A saddle point is stationary ($f_x=0, f_y=0$) but is neither a max nor a min.

💡 Intuition: At the constrained optimum, the contour lines of the objective function are tangent to the constraint boundary surface.

📐 Core Rule: Auxiliary function: $F = f + \lambda \phi$. Solve $\nabla f + \lambda \nabla \phi = 0$: $$f_x + \lambda \phi_x = 0, \quad f_y + \lambda \phi_y = 0, \quad f_z + \lambda \phi_z = 0$$ alongside the constraint $\phi = 0$.

🎯 Exam Cue: Eliminating $\lambda$ between ratios $\frac{f_x}{\phi_x} = \frac{f_y}{\phi_y} = \frac{f_z}{\phi_z}$ reveals symmetrical variable relations (e.g. $x=y=z$).

💡 Intuition: Rank is the fundamental dimension of linear information in a matrix (number of linearly independent rows or columns).

📐 Core Rule: 1. Row Echelon Form: Uses row operations only; produces staircase of leading non-zero pivots. 2. Normal Form: Uses both row AND column operations to reduce matrix into $\begin{bmatrix} I_r & 0 \\ 0 & 0 \end{bmatrix}$, where $r = \text{Rank}(A)$.

🎯 Exam Cue: Rank of non-zero matrix $A_{m \times n}$ satisfies $1 \le \text{Rank}(A) \le \min(m, n)$.

💡 Intuition: Compare the rank of the coefficient matrix $A$ against the augmented matrix $[A|B]$ (Rouché–Capelli theorem).

📐 Core Rule: 1. $\text{Rank}(A) \neq \text{Rank}(A|B) \implies$ Inconsistent (No solution) 2. $\text{Rank}(A) = \text{Rank}(A|B) = n \implies$ Consistent, Unique solution 3. $\text{Rank}(A) = \text{Rank}(A|B) = r < n \implies$ Consistent, Infinite solutions ($n-r$ free parameters)

🎯 Exam Cue: For homogeneous systems $AX = 0$: Always consistent. $\text{Rank}=n \implies$ trivial $X=0$; $\text{Rank}<n \implies$ non-trivial infinite solutions.

💡 Intuition: Every square matrix satisfies its own scalar characteristic polynomial: $|A - \lambda I| = 0 \implies A^n + c_{n-1}A^{n-1} + \dots + c_0 I = 0$.

📐 Core Rule: Multiply entire equation by $A^{-1}$: $$A^{-1} = -\frac{1}{c_0}\big[A^{n-1} + c_{n-1}A^{n-2} + \dots + c_1 I\big]$$

🎯 Exam Cue: Also allows computing high powers (e.g. $A^8$) by polynomial division: $A^8 = Q(A) \cdot P(A) + R(A) = 0 + R(A)$.