Multivariable Calculus

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In Mathematics, multivariable calculus or multivariate calculus is an extension of calculus in one variable with functions of several variables. The differentiation and integration process involves multiple variables, rather than once. Let us discuss the definition of multivariable calculus, basic concepts covered in multivariate calculus, applications and problems in this article.

What is Multivariable Calculus?

Multivariable Calculus deals with the functions of multiple variables, whereas single variable calculus deals with the function of one variable. The differentiation and integration process are similar to the single variable calculus. In multivariable calculus, to find a partial derivative, first, take the derivative of the appropriate variable while holding the other variables as constant. It majorly deals with three-dimensional objects or higher dimensions. The typical operations involved in the multivariable calculus are:

Limits and Continuity
Partial Differentiation
Multiple Integration

Multivariable Calculus Topics

The important topics covered in the multivariable calculus are as follows:

Multivariable Calculus Topics
1. Differential Calculus	Partial Derivatives Differentiation Derivatives Chain Rule Directional Derivative and the Gradient Applications of Differential Calculus
2. Integral calculus	Double integrals Triple Integrals Changing Variables
3. Curves and surfaces	Parametrized Curve Length of the Curve Parameterized Surfaces Surface Area of Parameterized Surfaces
4. Vector Field	Vector Field Basics Vector Operators
5. Integration over curves and Surfaces	Line Integral Surface Integral
6. Fundamental Theorem of Vector Calculus	Gradient Theorem for Line Integrals Green’s Theorem Stokes’ Theorem Divergence Theorem

Multivariable Calculus Applications

One of the core tools of Applied Mathematics is multivariable calculus. It is used in various fields such as Economics, Engineering, Physical Science, Computer Graphics, and so on. Some of the applications of multivariable calculus are as follows:

Multivariable Calculus provides a tool for dynamic systems.
It is used in a continuous-time dynamic system for optimal control.
In regression analysis, it helps to derive the formulas to estimate the relationship among the set of empirical data.
In Engineering and Social Science, it helps to study and model the high dimensional systems that exhibit the deterministic nature.
In Finance, Quantitative Analyst uses multivariable calculus to predict future trends in the stock market.

Multivariable Calculus Problems

The multivariable calculus basic problems are given below

Example 1:

Find the first partial derivative of the function z = f (x, y) = x³ + y⁴ + sin xy.

Solution:

Given Function: z = f (x, y) = x³ + y⁴ + sin xy.

For the given function, the first partial derivative with respect to x is:

\(\frac{\partial z}{\partial x}=\frac{\partial f}{\partial x} =3x^{2}+cos(xy)y\)

Similarly, the first partial derivative with respect to y is:

\(\frac{\partial z}{\partial y}=\frac{\partial f}{\partial y} =4y^{3}+cos(xy)x\)

Example 2:

Find the total differential of the function: z = 2x sin y – 3x²y².

Solution:

Given function: z = 2x sin y – 3x²y².

The total differentiation of the function is given as:

\(dz = \frac{\partial z}{\partial x}dx +\frac{\partial z}{\partial y}dy\) \(dz = (2 sin y -6xy^{2})dx + (2x cos y – 6x^{2}y)dy\)

Example 3:

Find dw/dt if w = x²y – y², x = sin t, and y = e^t using chain rule.

Solution:

Given:

w = x²y – y², x = sin t, and y = e^t.

To find: dw/dt

\(\frac{dw}{dt}= \frac{\partial w}{\partial x}\frac{dx}{dt}+\frac{\partial w}{\partial y}\frac{dy}{dt}\)

dw/dt = 2xy (cos t) + ( x² -2y)e^t

dw/dt = 2 (sin t)(e^t) cos t + (sin² t -2e^t)e^t.

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