Partial derivatives, multiple integrals, and vector calculus terms from a multivariable calculus course.
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- What does the partial derivative ∂f/∂x measure?
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The rate of change of f with respect to x while treating all other variables as constants.- What is the notation for the partial derivative of f with respect to x?
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∂f/∂x or f_x- What are mixed partial derivatives?
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Partial derivatives taken with respect to different variables in sequence, such as ∂²f/∂x∂y.- What does Clairaut's theorem state about mixed partial derivatives?
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If f_xy and f_yx are continuous, then f_xy = f_yx (the order of differentiation does not matter).- What is the second partial derivative test used for?
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Determining whether a critical point is a local maximum, local minimum, or saddle point.- State the chain rule for partial derivatives when z = f(x,y) and x and y are functions of t.
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dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)- What is the equation of the tangent plane to a surface z = f(x,y) at point (x0, y0, z0)?
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z - z0 = f_x(x0, y0)(x - x0) + f_y(x0, y0)(y - y0)- What is the linear approximation formula for f(x,y) near (a,b)?
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f(x,y) ≈ f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)- What is the directional derivative of f at point P in the direction of unit vector u?
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D_u f = ∇f · u, the dot product of the gradient with the unit vector.- What is the gradient vector ∇f for a function f(x,y,z)?
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∇f = (f_x, f_y, f_z), a vector of all partial derivatives.- In which direction does the gradient vector point?
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The direction of greatest increase of the function.- What is a critical point of a multivariable function?
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A point where all first partial derivatives are zero or undefined.- What is a saddle point on a surface?
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A critical point that is a local maximum in one direction and a local minimum in another direction.- How is a double integral defined as a limit?
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∬_R f(x,y) dA = lim (sum of f(x_ij*, y_ij*) ΔA_ij) as the partition size approaches zero.- What is Fubini's theorem for double integrals?
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∬_R f(x,y) dA = ∫∫ f(x,y) dy dx = ∫∫ f(x,y) dx dy (under appropriate conditions).
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