Multivariable Calc Study

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What is a mixed partial derivative?

A derivative taken with respect to different variables in sequence

What is Clairaut’s Theorem?

A theorem stating that if the mixed partial derivatives are continuous, then they are equal

What is a gradient vector?

A vector that points in the direction of the greatest rate of increase of a function.

What is a level curve?

A curve along which a function of two variables has a constant value.

What is the tangent plane to a surface?

A plane that best approximates the surface near a given point.

What is the directional derivative?

The rate of change of a function in the direction of a given vector.

What is a critical point?

A point where all partial derivatives of a function are zero or undefined.

What is the second derivative test used for in multivariable calculus?

To classify critical points as local minima, maxima, or saddle points.

What is a function of several variables?

A rule that assigns to each ordered n-tuple in a domain a unique real number output.

What is the domain of a multivariable function?

The set of all input values (tuples) for which the function is defined.

What is the definition of a limit for multivariable functions?

A limit exists if the value approaches the same number from every path toward a point.

What is continuity in multivariable calculus?

A function is continuous at a point if the limit exists and equals the function’s value there.

What is the gradient vector?

A vector of partial derivatives: ∇f=⟨fx,fy,fz⟩

What does the gradient vector indicate?

The direction of greatest increase of a function.

How is the directional derivative calculated?

∇f⋅u⃗ , where u⃗ is a unit vector in the desired direction.

What is the equation of a tangent plane at a point (a,b)?

z=f(a,b)+fx​(a,b)(x−a)+fy​(a,b)(y−b)

How do you find critical points?

Set all partial derivatives equal to zero and solve.

What is the method of Lagrange Multipliers used for?

Finding maxima and minima of a function subject to a constraint.

What equations are solved using Lagrange Multipliers?

∇f=λ∇g and g(x,y,z)=c

What is a double integral over a region R

∬R​ f(x,y) dA: The volume under a surface over R

What is Fubini’s Theorem?

It allows switching the order of integration for continuous functions over rectangular regions.

Why use a change of variables in multiple integrals?

To simplify the region or integrand using substitution.

What is the Jacobian determinant?

The determinant of the matrix of partial derivatives, used to adjust area/volume under substitution.

What is the area element in polar coordinates?

dA=r dr dθ

What is the volume element in cylindrical coordinates

dV=r dr dθ dz

What is the volume element in spherical coordinates?

dV= ρ^2 sin⁡ϕ dρ dϕ dθ

What is the divergence of a vector field F⃗?

∇⋅F, a scalar measuring outflow at a point.

What is the curl of a vector field F⃗?

∇×F, a vector describing rotation around a point.

What is Green’s Theorem?

Relates a line integral around a simple closed curve to a double integral over the region it encloses.

What is Stokes’ Theorem?

Relates a surface integral of curl F⃗ to a line integral of F⃗ over the boundary curve.

What is the Divergence Theorem?

Relates the flux of F⃗ across a closed surface to the triple integral of the divergence of F⃗.

What is a surface integral of a scalar function?

An integral of the form ∬Sf(x,y,z) dS, representing the total value of f over a surface.

What does the surface element dS represent?

The infinitesimal area element of a surface in 3D space.

How is dS computed for a parametric surface r⃗(u,v)?

dS=∥ru​×rv​∥du dv

What is a surface integral of a vector field?

∬S​F⋅dS, representing the flow of F⃗ through a surface S

What is dS⃗ in a vector surface integral?

The vector surface element: dS⃗=n⃗ dS, where n⃗ is a unit normal vector.

What is flux in vector calculus?

The quantity of a vector field passing through a surface.

What is the physical flow interpretation of a surface integral of a vector field?

It represents how much of the vector field passes through the surface—positive when going in the direction of the normal, negative otherwise.

How is orientation of the surface relevant in computing flux?

It determines the direction of the normal vector and thus affects the sign of the flux.