What is the geometric interpretation of the dot product u · v?

The product of the magnitudes of u and v multiplied by the cosine of the angle between them.

What does the Jacobian ∂(x,y)/∂(u,v) represent in a change of variables for double integrals?

The scaling factor for area when transforming from (u,v) to (x,y) coordinates.

MATH 2023: Vectors and Multivariable Calculus Essentials

This cheat sheet summarizes key concepts from MATH 2023, covering vectors, geometry in space, multivariable functions, calculus, and vector fields. It provides a concise overview of definitions, properties, and applications.

Core Principles

Vectors in R³: Represented by magnitude and direction, with operations like addition, subtraction, and scalar multiplication.
Lines and Planes: Defined parametrically and by equations, with methods to determine their relative positions and distances.
Multivariable Functions: Extend single-variable concepts to functions of multiple variables, visualized through graphs and level curves/surfaces.
Partial Derivatives: Measure the rate of change of a multivariable function with respect to one variable, holding others constant.
Multiple Integrals: Used to calculate volumes, areas, and other quantities over regions in 2D and 3D space, often simplified by coordinate transformations.
Vector Fields: Functions assigning a vector to each point in space, crucial for understanding concepts like work, flux, and conservative fields.

Formulas

Distance between points A=(x₀, y₀, z₀) and B=(x₁, y₁, z₁): |AB| = √((x₁-x₀)² + (y₁-y₀)² + (z₁-z₀)²)
Dot Product: u · v = |u||v|cos θ
Cross Product Magnitude: |u × v| = |u||v|sin θ
Line in Space (Parametric): γ(t) = (x₀, y₀, z₀) + t(a, b, c)
Plane Equation: ax + by + cz + d = 0
Distance from point P₀ to plane ax+by+cz+d=0: |ax₀+by₀+cz₀+d| / √(a²+b²+c²)
Parametric Curve: r(t) = (x(t), y(t), z(t))
Tangent Vector to curve: r'(t)
Partial Derivatives: fx(a, b) = lim(h→0) [f(a+h, b) - f(a, b)] / h
Tangent Plane to graph z=f(x,y) at (x₀,y₀,z₀): z - z₀ = fx(x₀, y₀)(x - x₀) + fy(x₀, y₀)(y - y₀)
Tangent Plane to level surface f(x,y,z)=k at (x₀,y₀,z₀): fx(x₀,y₀,z₀)(x-x₀) + fy(x₀,y₀,z₀)(y-y₀) + fz(x₀,y₀,z₀)(z-z₀) = 0
Linear Approximation: f(x, y) ≈ f(x₀, y₀) + fx(x₀, y₀)(x - x₀) + fy(x₀, y₀)(y - y₀)
Chain Rule (one independent variable): dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)
Directional Derivative: Duf(x₀, y₀) = ∇f(x₀, y₀) · u
Second Derivative Test: D = fxx fyy - (fxy)²
Lagrange Multipliers: ∇f = λ∇g
Double Integral (Fubini's Theorem): ∫∫ f(x,y)dA = ∫∫ f(x,y)dydx = ∫∫ f(x,y)dxdy
Jacobian for Polar Coordinates: ∂(x,y)/∂(r,θ) = r
Surface Area: ∫∫D √(fx² + fy² + 1) dA
Triple Integral (Fubini's Theorem): ∫∫∫ f(x,y,z)dV = ∫∫∫ f(x,y,z)dzdydx (and other orders)
Jacobian for Cylindrical Coordinates: ∂(x,y,z)/∂(r,θ,z) = r
Jacobian for Spherical Coordinates: ∂(x,y,z)/∂(ρ,θ,φ) = ρ²sinφ
Line Integral of Vector Field: ∫ F·dr = ∫ (Pdx + Qdy + Rdz)
Fundamental Theorem of Line Integrals: ∫ F·dr = f(r(b)) - f(r(a))
Curl of F: curl F = ∇ × F
Divergence of F: div F = ∇ · F

Quiz

What is the geometric interpretation of the dot product u · v?: The product of the magnitudes of u and v multiplied by the cosine of the angle between them.
Which of the following is NOT a condition for a vector field F = Pi + Qj to be conservative in a simply connected domain?: ∂P/∂x = ∂Q/∂y
What does the Jacobian ∂(x,y)/∂(u,v) represent in a change of variables for double integrals?: The scaling factor for area when transforming from (u,v) to (x,y) coordinates.