University of Lagos GEG 311 Calculus of Several Variables Exam Study Guide March 2024

Mathematical Definitions and Vector Field Theory

• Linear Dependence:     * A set of vectors $\{v_1, v_2, ..., v_n\}$ in a vector space $V$ is said to be linearly dependent if there exist scalars $a_1, a_2, ..., a_n$, not all zero, such that:     * $a_1 v_1 + a_2 v_2 + ... + a_n v_n = 0$     * In simple terms, at least one vector in the set can be expressed as a linear combination of the others.

• Linear Independence:     * A set of vectors $\{v_1, v_2, ..., v_n\}$ is linearly independent if the only solution to the vector equation:     * $a_1 v_1 + a_2 v_2 + ... + a_n v_n = 0$     * is the trivial solution, where all scalars are zero: $a_1 = a_2 = ... = a_n = 0$.

• Conservative Vector Fields:     * A vector field $\mathbf{F}$ is conservative if it is the gradient of some scalar function $\phi$ (the potential function).     * Mathematical expression: $\mathbf{F} = \nabla \nu$     * Properties: The line integral of a conservative field along a closed loop is zero, and the curl of the field is zero: $\nabla \times \text{F} = \text{0}$.

• Fundamental Theorem of Calculus (Multivariable context):     * This theorem relates the line integral over a curve to the values of a potential function at the endpoints of the curve.     * Mathematical expression: $\int_C \nabla f \times d\text{r} = f(r(b)) - f(r(a))$     * Where $C$ is a smooth curve from point $A$ to point $B$.

• Green's Theorem:     * Relates a line integral around a simple closed curve $C$ to a double integral over the plane region $D$ bounded by $C$.     * Mathematical expression: \oint_C (P dx + Q dy) = \text{\int\int}_D (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA     * Assumes $C$ is a positively oriented, piecewise smooth, simple closed curve.

Total Differentials and Higher Order Derivatives

• Given Function:     * $z = x^2 + 8xy + 3y^3$

• Total Differential Formula:     * The general form is: $dz = f(x, y) dx + g(x, y) dy$     * Partial derivative with respect to $x$ ($f(x, y)$): $\frac{\partial z}{\partial x} = 2x + 8y$     * Partial derivative with respect to $y$ ($g(x, y)$): $\frac{\partial z}{\partial y} = 8x + 9y^2$     * Result: $dz = (2x + 8y) dx + (8x + 9y^2) dy$

• Second Order Derivatives:     * $f_{xx} = \frac{\partial^2 z}{\partial x^2} = \frac{\partial}{\partial x}(2x + 8y) = 2$     * $f_{yy} = \frac{\partial^2 z}{\partial y^2} = \frac{\partial}{\partial y}(8x + 9y^2) = 18y$     * $f_{xy} = f_{yx} = \frac{\partial^2 z}{\partial x ∂ y} = \frac{\partial}{\partial x}(8x + 9y^2) = 8$

Linear Algebra and Vector Spaces

• Jacobian Test for Functional Dependence:     * System defined as:         * $y_1 = 4x_1 - x_2$         * $y_2 = x + 6x_2x + 9x$ (Note: Based on transcript text, which appears to have typographical inconsistencies regarding indices).     * The test involves finding the determinant of the Jacobian matrix $J = \frac{\partial(y_1, y_2)}{\partial(x_1, x_2)}$. If the determinant is zero, the functions are dependent.

• Bases in $P_2(ℝ)$:     * Vectors provided:         * $P_1(x) = 1 + x - 3x^2$         * $P_2(x) = 1 + 2x - 3x^2$         * $P_3(x) = -x + x^2$     * To prove they form a basis for the space of polynomials of degree $≤ 2$, one must show they are linearly independent (e.g., by checking the determinant of their coefficients) and that they span the 3-dimensional space.     * Linear combination task: Find scalars $c_1, c_2, c_3$ such that $P(x) = 2 + 3x - 3x^2 = c_1 P_1(x) + c_2 P_2(x) + c_3 P_3(x)$.

Multi-Variable Vector Functions and Jacobians

• Function definition:     * $F: ℝ^3 \rightarrow ℝ^2$ defined by $F(x, y, z) = (x^2 + 3y + z^3, xy + z^2 + 1)$.

• Jacobian Matrix Formulation:     * The matrix of first-order partial derivatives for $F = (f_1, f_2)$:     * $J = \begin{pmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} & \frac{\partial f_1}{\partial z} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} & \frac{\partial f_2}{\partial z} \end{pmatrix} = \begin{pmatrix} 2x & 3 & 3z^2 \\ y & x & 2z \end{pmatrix}$

• Critical Points and Critical Values:     * Critical points occur where the Jacobian matrix has rank less than the dimension of the codomain (i.e., less than 2).

Green's Theorem Applications

• Problem: Compute $\oint_{∂ D} x^4 dx + xydy$ where $D$ is the triangular region with vertices $(0,0), (1,0), (0,1)$.

• Parameters:     * $P = x^4$     * $Q = xy$     * $\frac{\partial Q}{\partial x} = y$     * $\frac{\partial P}{\partial y} = 0$

• Double Integral Conversion:     * $\iint_D (y - 0) dA = \int_0^1 \int_0^{1-x} y dy dx$     * Calculation: Evaluate the inner integral $\int_0^{1-x} y dy = [\frac{y^2}{2}]_0^{1-x} = \frac{(1-x)^2}{2}$. Then evaluate $\int_0^1 \frac{(1-x)^2}{2} dx$.

Equations of Planes and Lines

• Problem: Find an equation $ax + by + cz = d$ for the plane passing through $(-2, -1, 4)$ perpendicular to a given line.

• Parametric Equations of the Line:     * $x = 2t$     * $y = 3t - 1$     * $z = 5 - t$

• Normal Vector determination:     * The direction vector of the line is $\mathbf{v} = (2, 3, -1)$.     * Since the plane is perpendicular to the line, this direction vector is the normal vector $\mathbf{n}$ of the plane.

• Equation derivation:     * $2(x - (-2)) + 3(y - (-1)) - 1(z - 4) = 0$     * $2(x + 2) + 3(y + 1) - (z - 4) = 0$     * $2x + 4 + 3y + 3 - z + 4 = 0 \rightarrow 2x + 3y - z = -11$

Analysis of Continuity and Partial Derivatives

• Piecewise Function:     * $f(x, y) = \frac{3x^3 - 5y^3}{x^2 + y^2}$ for $(x, y) \neq (0,0)$, and $f(0,0) = 0$.

• Partial Derivatives at the origin:     * $f_x(0,0) = ℒ_{h \rightarrow 0} \frac{f(h, 0) - f(0, 0)}{h} = \frac{\frac{3h^3}{h^2} - 0}{h} = 3$     * $f_y(0,0) = ℒ_{h \rightarrow 0} \frac{f(0, h) - f(0, 0)}{h} = \frac{\frac{-5h^3}{h^2} - 0}{h} = -5$

• Continuity Evaluation:     * A function is continuous at $(0,0)$ if the limit as $(x,y) \rightarrow (0,0)$ equals the function value ($0$).     * Using polar coordinates ($x = r \cos(\theta)$, $y = r \sin(\theta)$):     * $ℒ_{r \rightarrow 0} \frac{3r^3 \cos^3(\theta) - 5r^3 \sin^3(\theta)}{r^2} = ℒ_{r \rightarrow 0} r(3 \cos^3(\theta) - 5 \sin^3(\theta)) = 0$     * Since the limit is $0$ regardless of the path (the angle $\theta$), the function is continuous.

Advanced Theorems and Transformations

• Stokes' Theorem:     * Relates the line integral of a vector field along the boundary of surface $S$ to the surface integral of the curl of the field.     * \oint_C \text{F} \cdot d\text{r} = \text{\int\int}_S (\nabla \times \text{F}) \cdot d\text{S}

• Divergence (Gauss's) Theorem:     * Relates the flux of a vector field through a closed surface $S$ to the volume integral of the divergence of the field over the region $V$ enclosed by $S$.     * \text{\int\int}_S \text{F} \cdot d\text{S} = \text{\int\int\int}_V (\nabla \times \text{F}) dV

• Kernel (Null Space) of a Linear Transformation:     * The kernel of $T: V \rightarrow W$ is the set of all vectors $v$ in $V$ such that $T(v) = 0$.     * To prove the kernel is a subspace, show that it contains the zero vector and is closed under addition ($v_1 + v_2$) and scalar multiplication ($kv$).

• Nullity of $T: V \rightarrow ℝ$:     * Where $V$ is the space of polynomials of degree $\u2264 2$ ($dim(V) = 3$).     * According to Rank-Nullity Theorem: $dim(V) = rank(T) + nullity(T)$.     * Since the range is $\u211D$, the rank can be $0$ (if $T(v)=0$ for all $v$) or $1$ (if $T$ is onto).     * Possible nullities are $2$ or $3$.

Variational Calculus and Gradients

• Euler Equation for Functional Optimization:     * Given functional: $I[y] = \int (y^2 + 4yy' + 4(y')^2) dx$     * Equation used: $\frac{\partial F}{\partial y} - \frac{d}{dx}(\frac{\partial F}{\partial y'}) = 0$     * Where $F(x, y, y') = y^2 + 4yy' + 4(y')^2$

• Directional Gradients and Normals:     * Unit Normal Vector: For surface $x^3 + y^3 + z = 4$ at point $(1, 1, 2)$.         * $\nabla f = (3x^2, 3y^2, 1)$         * At $(1, 1, 2)$, $\nabla f = (3, 3, 1)$         * Unit vector: $\frac{(3, 3, 1)}{\sqrt{3^2 + 3^2 + 1^2}} = \frac{(3, 3, 1)}{\sqrt{19}} = (\frac{3}{\sqrt{19}}, \frac{3}{\sqrt{19}}, \frac{1}{\sqrt{19}})$     * Greatest Rate of Increase: The direction of greatest increase of $f$ is the direction of the gradient vector $\nabla f$.         * For $f(x, y, z) = xye^z$ at $(2, 1, 2)$.

• Gradient Vector Field Visualization:     * For $f(x, y) = x^2 + y^2$:     * Gradient field: $\nabla f = (2x, 2y)$. Vectors point radially outward from the origin, increasing in magnitude as distance from the origin increases.     * Contours: Concentric circles centered at the origin defined by $x^2 + y^2 = k$.

Tensor Analysis Identity

• Identity to Prove:     * $\nabla ∙ (ϕ \text{v}) = ϕ (\nabla ∙ \text{v}) + \text{v} ∙ (\nabla ϕ)$     * Proof via Tensor Analysis: Using standard index notation or basis vectors ($HINT: A = ae_i, B = be_j$).