Intuition The one core ideaWhen you re-label the points of a plane with new coordinates, tiny patches of area get stretched or squeezed by a number that depends on where you are. To keep an integral honest, you must multiply by that local stretch factor — and that factor is the absolute value of the Jacobian determinant .
This page builds every symbol the parent note Change of Variables leans on, starting from nothing. If a word or squiggle appears there and confuses you, it is explained below, in the order that lets each idea rest on the previous one.
Everything here lives in two flat planes side by side.
The uv choose nice coordinates.
The x y
Definition A coordinate pair( u , v )
We will constantly move a point from the left sheet to the right sheet. That move is called a map .
Definition Map / transformationT rule that takes each point ( u , v ) ( x , y ) T ( u , v ) = ( x , y )
The arrow ↦ gets sent to ". So ( u , v ) ↦ ( x , y )
The colon in T : ( u , v ) ↦ ( x , y ) is the rule that ".
Concretely a map is really two ordinary functions bundled together :
x = x ( u , v ) , y = y ( u , v ) . x ( u , v ) x u v
Intuition Why we need a mapWe want to swap coordinates inside an integral. The map is the exact dictionary translating every new point into where it lands in the old world. Without it we could not rewrite the integrand or the region.
Two words the parent attaches to T
The map has no rips, jumps, or sharp corners — you can take derivatives of x ( u , v ) y ( u , v )
Definition Injective (one-to-one)No two different new points land on the same old point. Picture: the map never folds the sheet back onto itself. This matters because if two cells piled onto the same spot, we'd count that spot's area twice.
The parent writes r ( u 0 , v 0 ) r u r
Definition Position vectorr r ( u , v ) = ( x ( u , v ) , y ( u , v )) arrow from the origin to that point. Bold letters mean "this is a vector (an arrow), not a single number."
Why bother turning a point into an arrow? Because arrows can be subtracted to get displacement arrows, and displacements are exactly what we need to measure the size of a mapped patch.
This is the tool that measures how the output moves when you nudge one input .
partial derivative, not an ordinary one?
Our output x two inputs u v the input changes" — but we have two inputs. So we ask a sharper question: "if I freeze v u x partial derivative .
Definition Partial derivative notation∂ u ∂ x = x u = lim h → 0 h x ( u + h , v ) − x ( u , v ) .
The curly ∂ d there are other variables held constant ".
The subscript form x u
Plain words: the rate x u v
Definition The velocity vectorsr u r v r u = ( x u , y u ) , r v = ( x v , y v ) . Picture: r u per unit step in u . It points along the direction the image moves when you slide right on the u r v v
Intuition Why these two arrows are the whole gameA tiny cell in the uv u v r u d u r v d v
The link Chain Rule (Multivariable) is where these partials get combined when maps are composed; the reciprocal trick in the parent leans on it.
Definition The differentialsd u , d v d u infinitely small width along u d v v
Definition The area elementd A d A = d x d y tiny area of one microscopic patch in the x y d A not equal to d u d v
Why "infinitely small"? Because only when a patch is small does the smooth map look straight (linear) across it. That straightness is what lets a triangle-and-parallelogram argument work.
The mapped patch is a parallelogram (a slanted rectangle). We need its area.
Definition Parallelogram spanned by two arrowsGiven edge-arrows a b
Intuition Why this exact combination measures areaThe quantity a x b y − a y b x cross product — see Cross Product and Area . It equals ∣ a ∣ ∣ b ∣ sin γ γ × = ∣ a ∣ × ( ∣ b ∣ sin γ )
Definition Why the answer can come out negative before the barsIf the two edges are swapped (or the map flips the sheet like a mirror), a x b y − a y b x negative . Area is never negative, so we wrap it in ∣ ⋯ ∣
The parent bundles those four partials into a matrix and takes its determinant . Both symbols are new; here they are.
2 × 2 ( p r q s ) columns : column one is r u r v
Definition Determinant of a2 × 2 det ( p r q s ) = p s − q r . det p s − q r exactly the cross-product formula from §5 — because the columns are the two edge-arrows.
Intuition Why the determinantis the stretch factor
Feed the determinant the unit square (edges ( 1 , 0 ) ( 0 , 1 ) det Determinant as Volume Scaling . A negative determinant = the map turned the sheet over (mirror flip); its size is still the stretch.
Now every piece assembles into the star of the topic.
Definition Jacobian determinant∂ ( u , v ) ∂ ( x , y ) = det ( x u y u x v y v ) = x u y v − x v y u .
The stacked notation ∂ ( u , v ) ∂ ( x , y ) one symbol meaning "the determinant of the grid of all partials of ( x , y ) ( u , v ) Top = output variables, bottom = input variables.
Plain words: the local area-stretch of the map T
Common mistake Reading the fraction as a normal division∂ ( u , v ) ∂ ( x , y ) not a numerator over a denominator you can cancel. It is a whole determinant. The only fraction-like fact allowed is the reciprocal rule with the inverse map — which comes from the Inverse Function Theorem .
The absolute value ∂ ( u , v ) ∂ ( x , y )
Definition Double integral∬ R f d A R f d A ∫ R which region you sum over. See Double Integrals over General Regions .
R S
S new (uv R old (x y R = T ( S ) ∬ R ( old ) = ∬ S ( new ) × ∣ J ∣ both the integrand and the region.
partial derivatives x_u y_v
velocity edge arrows r_u and r_v
parallelogram area via cross product
determinant as area scaling
Jacobian dx dy over du dv
double integral over a region
change of variables theorem
Test yourself — say the answer aloud before revealing.
What does the arrow ↦ ( u , v ) ↦ ( x , y ) "gets sent to" — the input point on the left produces the output point on the right.
What does "injective" guarantee about the map? No two different points land on the same output, so the sheet never folds onto itself and no area is double-counted.
What is x u The rate at which x u v
Why is ∂ d To flag that other variables are being held constant (there is more than one input).
What geometric object are r u r v Arrows showing how fast and in what direction the output moves per unit step in u v
What shape does a tiny uv A parallelogram (to first order), spanned by r u d u r v d v
What does a x b y − a y b x The signed area of the parallelogram framed by arrows a b
Why take an absolute value of that quantity? Its sign only records orientation (clockwise vs counter-clockwise / a mirror flip); area itself must be non-negative.
What does the determinant of a 2 × 2 The factor by which the linear map scales area.
In ∂ ( u , v ) ∂ ( x , y ) The output (old) variables x , y u , v
What is d A d u d v The tiny old-plane area d x d y d A = ∣ J ∣ d u d v
What must you translate when changing variables, besides the integrand? The region — the new limits S uv
