Multivariable Calculus

Multivariable Calculus explores functions of multiple variables, including partial derivatives, multiple integrals, and vector calculus.

Functions of Several VariablesPartial DerivativesMultiple Integrals
Gradient, Divergence, and CurlLine and Surface IntegralsTheorems of Vector Calculus
Modeling Heat and DiffusionOptimizing Engineering DesignsData Science and Gradient Descent
Visualizing Surfaces and RegionsBreaking Problems into Steps
Functions of Several...Partial DerivativesMultiple IntegralsGradient, Divergence...Line and Surface Int...Theorems of Vector C...Modeling Heat and Di...Optimizing Engineeri...Data Science and Gra...Visualizing Surfaces...Breaking Problems in...
Basic Concepts

Multiple Integrals

Adding Up Over Regions

Multiple integrals let us sum or accumulate values over areas (double integrals) or volumes (triple integrals). They're the multivariable equivalent of adding up slices in single-variable calculus.

Double Integrals

A double integral \(\iint_R f(x, y),dx,dy\) calculates things like total mass, area, or charge over a region \(R\).

Triple Integrals

Triple integrals \(\iiint_D f(x, y, z),dx,dy,dz\) allow us to find volumes or total quantities in 3D.

Setting Up the Integral

  • Define the region of integration.
  • Set the order of integration (dx, dy, dz).
  • Integrate step by step.

Applications

These integrals are powerful for calculating things in physics, engineering, and probability.

Examples

  • Finding the total rainfall over a city by integrating rainfall intensity across the city's area.

  • Calculating the mass of a metal block with varying density using a triple integral.

In a Nutshell

Multiple integrals let us add up values over areas and volumes, extending integration to higher dimensions.

Key Terms

Double Integral
An integral used to sum a function over a two-dimensional region.
Triple Integral
An integral used to sum a function over a three-dimensional region.
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Gradient, Divergence, and Curl