Multivariable Calculus Notes for BS

Multivariable calculus is a branch of calculus that extends the concepts of differential and integral calculus to functions of more than one variable.

Functions of Several Variables:

  • Functions of several variables are functions that depend on more than one independent variable. They denoted by ( f(x, y) ), ( g(x, y, z) ), or more generally ( f(x_1, x_2, \ldots, x_n) ).

Partial Derivatives:

  • A partial derivative of a function of several variables is the derivative of the function with respect to one of its variables, with the others held constant.


  • The gradient of a scalar-valued function ( f(x, y, z) ) is a vector that points in the direction of the greatest rate of increase of the function. It is denoted by ( \nabla f ):
    [ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) ]

Directional Derivative:

  • The directional derivative of a function ( f(x, y, z) ) in the direction of a unit vector ( \mathbf{v} = \langle a, b, c \rangle ) is given by:
    [ D_{\mathbf{v}} f = \nabla f \cdot \mathbf{v} = \frac{\partial f}{\partial x}a + \frac{\partial f}{\partial y}b + \frac{\partial f}{\partial z}c ]

Tangent Plane and Normal Line:

  • The equation of the tangent plane to the surface defined by ( z = f(x, y) ) at the point ( (x_0, y_0, z_0) ) is:
    [ z – z_0 = \frac{\partial f}{\partial x}(x – x_0) + \frac{\partial f}{\partial y}(y – y_0) ]
  • The normal line to the surface at the point ( (x_0, y_0, z_0) ) is parallel to the gradient ( \nabla f ).

Double and Triple Integrals:

  • Double integrals are used to integrate a function of two variables over a region in the plane. Triple integrals extend this concept to functions of three variables integrated over a region in space.

Change of Variables in Multiple Integrals:

  • The change of variables formula in multiple integrals allows one to integrate over a region by transforming the variables using a suitable transformation.

Line Integrals:

  • Line integrals are integrals along curves. They can calculate quantities such as work, circulation, and flux.

Surface Integrals:

  • Surface integrals are integrals over a surface. They can calculate quantities such as flux and surface area.

Vector Fields:

A vector field assigns a vector to each point in space. Examples include velocity fields, force fields, and electromagnetic fields.

These are some of the fundamental concepts and techniques in multivariable calculus, which are essential for understanding and solving problems in physics, engineering, and other fields involving functions of several variables.

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