Multivariable Calculus on the Subject GRE

GRE Math Subject · Axiom Academy

LESSON Multivariable Calculus on the Subject GRE Partial derivatives, multiple integrals, Jacobians, and the big integral theorems at statement level Step 1: Partial Derivatives and the Gradient Partial derivative: For , the partial derivative with respect to at is: Compute it by treating all other variables as constants and differentiating with respect to . The gradient points in the direction of steepest ascent and is perpendicular to level sets . Directional derivative: The rate of change of in the direction of a unit vector is: Maximum value: , achieved when . GRE Quick Hit: Given , find the directional derivative at in the direction of . Gradient: . At : . Unit vector: . Answer: . Step 2: Differentiability and the Chain Rule Total derivative (Jacobian matrix): A function is differentiable at if there exists a linear map such that: The matrix of is the Jacobian matrix : . GRE Trap: The existence of all partial derivatives does NOT imply differentiability in several variables. However, if all partial derivatives exist and are continuous in a neighborhood, then is differentiable there (this is the condition). Multivariable Chain Rule: If and , then: In matrix terms, Jacobians multiply: . Classic GRE setup: If and , find . Step 3: Multiple Integrals and Change of Variables Double integral: computes the "signed volume" under the surface over the region . Iterated integrals (Fubini's theorem): If is continuous on the rectangle , then:

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