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Multivariable Calculus Basics cheat sheet - grade 12

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Calculus Grade 12

Multivariable Calculus Basics Cheat Sheet

A printable reference covering multivariable functions, partial derivatives, gradients, tangent planes, double integrals, and polar coordinates for grade 12.

TopicsFunctionsDerivativesIntegralsVectorsTrigonometryCalculusStudy Guide
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Multivariable calculus studies functions that depend on two or more input variables, such as f(x,y)f(x,y) or f(x,y,z)f(x,y,z). Students need this cheat sheet to organize the main ideas that extend single-variable calculus into higher dimensions. It is especially useful for reviewing surfaces, rates of change, optimization, and accumulation over regions.

These tools appear in physics, engineering, economics, and advanced mathematics.

The core ideas are partial derivatives, gradients, tangent planes, and multiple integrals. A partial derivative measures how a function changes when one variable changes and the others stay fixed. The gradient ∇f\nabla f points in the direction of greatest increase and gives directional derivatives.

Double integrals such as ∬Rf(x,y) dA\iint_R f(x,y)\,dA add values over a two-dimensional region, often using rectangular or polar coordinates.

Key Facts

  • For a function z=f(x,y)z=f(x,y), the partial derivative with respect to xx is fx(x,y)=∂f∂x=lim⁡h→0f(x+h,y)−f(x,y)hf_x(x,y)=\frac{\partial f}{\partial x}=\lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h}.
  • For a function z=f(x,y)z=f(x,y), the partial derivative with respect to yy is fy(x,y)=∂f∂y=lim⁡h→0f(x,y+h)−f(x,y)hf_y(x,y)=\frac{\partial f}{\partial y}=\lim_{h\to 0}\frac{f(x,y+h)-f(x,y)}{h}.
  • The gradient of f(x,y)f(x,y) is ∇f=⟨fx,fy⟩\nabla f=\langle f_x,f_y\rangle, and for f(x,y,z)f(x,y,z) it is ∇f=⟨fx,fy,fz⟩\nabla f=\langle f_x,f_y,f_z\rangle.
  • The directional derivative of ff in the unit direction u\mathbf{u} is Duf=∇f⋅uD_{\mathbf{u}}f=\nabla f\cdot \mathbf{u}.
  • The tangent plane to z=f(x,y)z=f(x,y) at (a,b,f(a,b))(a,b,f(a,b)) is z=f(a,b)+fx(a,b)(x−a)+fy(a,b)(y−b)z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b).
  • A double integral over a rectangular region R=[a,b]×[c,d]R=[a,b]\times[c,d] can be written as ∬Rf(x,y) dA=∫ab∫cdf(x,y) dy dx\iint_R f(x,y)\,dA=\int_a^b\int_c^d f(x,y)\,dy\,dx.
  • In polar coordinates, x=rcos⁡θx=r\cos\theta, y=rsin⁡θy=r\sin\theta, and the area element becomes dA=r dr dθdA=r\,dr\,d\theta.
  • For critical points of f(x,y)f(x,y), solve fx=0f_x=0 and fy=0f_y=0, then use D=fxxfyy−(fxy)2D=f_{xx}f_{yy}-(f_{xy})^2 to classify local behavior when possible.

Vocabulary

Multivariable function
A function such as f(x,y)f(x,y) or f(x,y,z)f(x,y,z) that has more than one independent input variable.
Partial derivative
A derivative like ∂f∂x\frac{\partial f}{\partial x} that measures change in one variable while holding the other variables constant.
Gradient
The vector ∇f\nabla f made from the partial derivatives of ff, pointing in the direction of greatest increase.
Directional derivative
The rate of change of ff in a chosen unit vector direction u\mathbf{u}, given by Duf=∇f⋅uD_{\mathbf{u}}f=\nabla f\cdot\mathbf{u}.
Tangent plane
A plane that locally approximates a surface z=f(x,y)z=f(x,y) near a point using fxf_x and fyf_y.
Double integral
An integral ∬Rf(x,y) dA\iint_R f(x,y)\,dA that accumulates values of a function over a two-dimensional region RR.

Common Mistakes to Avoid

  • Differentiating both variables in a partial derivative, which is wrong because ∂f∂x\frac{\partial f}{\partial x} treats yy as a constant and ∂f∂y\frac{\partial f}{\partial y} treats xx as a constant.
  • Using a non-unit vector in Duf=∇f⋅uD_{\mathbf{u}}f=\nabla f\cdot\mathbf{u}, which gives the wrong rate because the formula requires u\mathbf{u} to have length 11.
  • Forgetting the factor rr in polar integrals, which is wrong because the area element changes from dAdA to r dr dθr\,dr\,d\theta.
  • Mixing the order and limits in an iterated integral, which can describe the wrong region if the bounds do not match the chosen order of integration.
  • Assuming fx=0f_x=0 and fy=0f_y=0 always gives a maximum or minimum, which is wrong because a critical point can also be a saddle point.

Practice Questions

  1. 1 Find fxf_x and fyf_y for f(x,y)=3x2y−4xy3f(x,y)=3x^2y-4xy^3, then evaluate both at (1,2)(1,2).
  2. 2 Find the tangent plane to z=x2+xy+y2z=x^2+xy+y^2 at the point (1,2,7)(1,2,7).
  3. 3 Evaluate ∫02∫03(x+y) dy dx\int_0^2\int_0^3 (x+y)\,dy\,dx.
  4. 4 Explain why the gradient ∇f(a,b)\nabla f(a,b) is perpendicular to the level curve f(x,y)=cf(x,y)=c at the point (a,b)(a,b).

Understanding Multivariable Calculus Basics

A useful way to picture a function of two inputs is as a landscape. Each location on a map has two coordinates, while the function value gives its height. Holding one input fixed means walking along a straight east to west or north to south path across that landscape.

The slope measured on each path can be different. This matters because a surface can rise sharply in one direction while staying nearly flat in another. When taking a partial derivative, students should identify which variable is changing before doing any algebra.

Treat every other variable as a constant number. This simple habit prevents many common errors.

Near a chosen point, a curved surface often behaves almost like a flat sheet. That local sheet is the tangent plane. It gives a quick estimate of nearby values without recalculating the full function.

Engineers use similar local estimates when an exact model is too complicated to work with directly. The estimate is most reliable close to the point of contact. It becomes less reliable farther away because curvature starts to matter.

The two partial derivatives control the tilt of the plane in two independent directions. A larger magnitude means a steeper tilt. Always check that the plane passes through the original point, since this catches setup mistakes.

Optimization in several variables is more subtle than finding a place where both first partial derivatives vanish. Such a point may be a hilltop, a valley floor, or a saddle point. A saddle rises along one path and falls along another, like a mountain pass.

The second partial derivatives describe the local bending of the surface. Their combined test can often sort the possible cases, but a zero result means the test gives no decision. Then students need another method, such as checking values along different paths or studying the function more closely.

Real design problems often include boundaries. The best value may occur on an edge or corner rather than at an interior critical point, so boundary checks are essential.

Double integrals build a total from many tiny pieces of area. If the function represents surface density, the total can represent mass. If it represents rainfall rate across land, the total can represent the amount of water collected.

The order of integration is mainly a choice of how to slice the region. Sketching the region first shows the correct limits and reveals whether one order is easier. Circular regions are usually simpler in polar coordinates.

The extra radius factor in the area element is necessary because narrow angular slices get wider as they move away from the origin. Forgetting that factor gives the wrong total. Good work in multivariable calculus depends as much on a clear sketch and sensible limits as on accurate differentiation.