Partial Derivatives and Local Change
Cheat sheet.

The key rules, formulas and reminders from all 4 topics, gathered into reference cards.

Key formulas, conditions and traps · Read down each column.

Surfaces, Level Sets and Multivariable Limits

Two paths with different limits disprove a multivariable limit. A bound in the distance to the target can prove an all-path limit.

Level curves lie in the input plane. Agreement on axes, or even all lines, is not an all-path proof.

L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b).

Partials alone do not imply continuity or differentiability. Continuous first partials nearby are a sufficient condition for a valid linearization.

D_uf=\nabla f\cdot u\quad(\|u\|=1),

\max_{\|u\|=1}D_uf=\|\nabla f\|.

Normalize directions. A zero gradient gives no preferred first-order direction and does not imply the function is locally constant.

\frac d{dt}f(x(t),y(t),t)=f_xx'+f_yy'+f_t,

z_x=-F_x/F_z.

Include every dependency and evaluate at the composed point. Implicit differentiation as a local graph needs the solved-for partial to be nonzero.

2 reference blocks

Two paths with different limits disprove a multivariable limit. A bound in the distance to the target can prove an all-path limit.

Level curves lie in the input plane. Agreement on axes, or even all lines, is not an all-path proof.

2 reference blocks

L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b).

Partials alone do not imply continuity or differentiability. Continuous first partials nearby are a sufficient condition for a valid linearization.

2 reference blocks

D_uf=\nabla f\cdot u\quad(\|u\|=1),

\max_{\|u\|=1}D_uf=\|\nabla f\|.

Normalize directions. A zero gradient gives no preferred first-order direction and does not imply the function is locally constant.

2 reference blocks

\frac d{dt}f(x(t),y(t),t)=f_xx'+f_yy'+f_t,

z_x=-F_x/F_z.

Include every dependency and evaluate at the composed point. Implicit differentiation as a local graph needs the solved-for partial to be nonzero.