Partial derivatives, gradients, Lagrange multipliers, double and triple integrals and coordinate changes.
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- Define the partial derivative of f with respect to x at point (a, b).
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The limit as h approaches 0 of [f(a+h,b) - f(a,b)] / h, denoted ∂f/∂x or f_x.- Define the partial derivative of f with respect to y at point (a, b).
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The limit as h approaches 0 of [f(a,b+h) - f(a,b)] / h, denoted ∂f/∂y or f_y.- What is the notation ∂f/∂x called?
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The partial derivative symbol, read as 'partial f partial x' or 'd round f d round x'.- State Clairaut's theorem (Schwarz's theorem).
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If f_xy and f_yx are continuous, then f_xy = f_yx (mixed partial derivatives are equal).- Write the chain rule for z = f(x(t), y(t)).
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dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt).- How do you find critical points of a function f(x, y)?
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Solve the system: ∂f/∂x = 0 and ∂f/∂y = 0.- Define the gradient vector of f(x, y).
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The vector grad f = (∂f/∂x, ∂f/∂y), or equivalently ∇f = (f_x, f_y).- What symbol represents the gradient operator?
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The nabla symbol, written as ∇ (an inverted triangle).- In which direction does the gradient vector point?
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In the direction of steepest ascent of the function.- Write the formula for the directional derivative of f at a point in direction of unit vector u.
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D_u f = ∇f · u = (∂f/∂x)u_1 + (∂f/∂y)u_2.- What is the maximum value of the directional derivative D_u f?
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The magnitude of the gradient vector, |∇f|.- What is the purpose of the method of Lagrange multipliers?
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To find extrema (maximum or minimum) of a function subject to a constraint.- Write the Lagrange function L for minimizing f(x, y) subject to g(x, y) = 0.
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L(x, y, λ) = f(x, y) - λ g(x, y).- State the critical point conditions for Lagrange multipliers with one constraint.
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∇f = λ ∇g, or equivalently ∂L/∂x = 0, ∂L/∂y = 0, ∂L/∂λ = 0.- How do you set up the Lagrange method with two constraints g(x, y, z) = 0 and h(x, y, z) = 0?
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Form L = f(x, y, z) - λ g(x, y, z) - μ h(x, y, z), then solve ∇L = 0.
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