Multivariable Calculus

Introduction

• In engineering, we come across various systems where the process depends on more than one variable.

• In this case, the optimization and designing of the system need the involvement of multiple variables.

• In this direction, multivariable calculus plays an important role. It has various applications starting from economics to science.

• In this tutorial, we will learn about multivariable calculus and its various operations (such as limits, continuity, partial derivatives, and integration) with solved examples.

What is meant by Multivariable Calculus

• Multivariable calculus is an extended topic of calculus that includes multiple variable functions.

• All the calculus operations are carried out on several variables rather than single variables.

• It is also considered the elementary part of advanced calculus.

  • It is extensively used to design and optimize dynamic systems. Moreover, it is applied for regression analysis in finance and engineering.

Limits and Continuity of Multivariable Functions

Limit

• We know that the limit of a function for a single variable is defined as the variable y approaches to b as the function g(y) approaches to a value P. Mathematically, it can be represented as $\mathrm{\displaystyle \lim \limits_{y \rightarrow b}g(y)=P}$

• Similarly, for multiple variables, limit is defined as the variable x,y approaches to (x₀,y₀) as the function g(y) approaches to a value P. Mathematically, it can be represented as $\mathrm{\displaystyle \lim \limits_{x,y \rightarrow x_0,y_0}g(y)=P}$.

Continuity

• Continuity is a fundamental concept of calculus.

• It is defined as the function so that a continuous variation of the argument induces a continuous variation of the values of the function.

• Continuity of multivariable function can be expressed as similar to single- variable function.

Mathematically, a function (g(y)) is said to be continuous if and only if it satisfies the following conditions.

• The function g(y) can be defined.

• The function g(y) should be continuous at (x₀,y₀) if $\mathrm{\displaystyle \lim \limits_{x,y \rightarrow x_0,y_0}g(y)=g(x_0,y_0)}$.

Partial Derivatives

• In calculus, the partial derivative is the derivative of a function containing multiple variables with respect to one variable at a time and other variables held constant.

• It is used in vector and differential calculus.

• Let’s consider a multivariable function g(x,y,z,....). The partial derivative of g with respect to x can be expressed as $\mathrm{g_x,g_x^ⵏ,∂_x g,D_x g,D_1 g,or\frac{\partial g}{\partial x}.}$

Now, we will see the formula for the partial derivative of a function g(x,y).

Partial derivative formula −

$\mathrm{g_x =\frac{\partial g}{\partial x}=\displaystyle \lim \limits_{h \rightarrow 0}\frac{g(x+h,y)-g(x,y)}{h} }$ (partial derivative of g(x,y) with respect to x)

Similarly, $\mathrm{g_y =\frac{\partial g}{\partial y}=\displaystyle \lim \limits_{h \rightarrow 0}\frac{g(x,y+h)-g(x,y)}{h} }$ (partial derivative of g(x,y) with respect to y)

Like differentiation, there are four types of rules used in partial derivatives.

• Product rule

• Quotient rule

• Chain rule

• Power rule

Multivariable Integration

In multivariable calculus, the multivariable integration is a definite integral of a function containing several variables. The integration of a function g(x,y)over a region R is called double integral.

Mathematically it can be represented as

$$\mathrm{\int\int_{R} g(x,y)dx dy}$$

Similarly, the integration of a function g(x,y,z)over a region R is called triple integral.

Mathematically it can be represented as

$$\mathrm{\int\int_{R} g(x,y,z)dx\: dy\: dz}$$

Partial Differential Equations

• The partial differential equations are the unique classification of differential equations.

• It is defined as the equation containing partial differentiation of a multivariable function.

• In other words, it establishes a relationship between the various partial derivatives of a multivariable function.

• It is abbreviated as PDE. The PDE for a function g(x,y,z,...) can be expressed as $\mathrm{\mathit{f}(x,y,....g,\frac{\partial g}{\partial x},\frac{\partial g}{\partial y},...,\frac{\partial^2 g}{\partial x \: \partial x},\dotsm\dotsm)=0.}$

Moreover, the partial derivatives of function g(x,y,z,...) can be represented as

$$\mathrm{g_x=\frac{\partial g}{\partial x}}$$

$$\mathrm{g_{xx}=\frac{\partial^2 g}{\partial x^2}}$$

$$\mathrm{g_{xy}=\frac{\partial^2 g}{\partial x\: \partial x}=\frac{\partial }{\partial y}(\frac{\partial g}{\partial x})}$$

There are various types of PDEs are used in mathematics such as

• First-order PDE

• Linear PDE

• Quasi-Linear PDE

• Homogeneous PDE

Solved Examples

Example 1:

Find the partial derivative of the following function with respect to x, y, and z.

$$\mathrm{g=5x^2+2y^2-6z^3-7xyz}$$

Solution:

It is given that,

$$\mathrm{g(x,y,z)=5x^2+2y^2-6z^3-7xyz}$$

$$\mathrm{Now, \frac{\partial g}{\partial x}=\frac{\partial }{\partial x}(5x^2+2y^2-6z^3-7xyz)}$$

$$\mathrm{\Rightarrow \frac{\partial g}{\partial x}=10x+0-0-7yz (y\: and\: z\: held\: constant)}$$

$$\mathrm{\Rightarrow \frac{\partial g}{\partial x}=10x-7yz}$$

$$\mathrm{\frac{\partial g}{\partial y}=\frac{\partial }{\partial y}(5x^2+2y^2-6z^3-7xyz)}$$

$$\mathrm{\Rightarrow \frac{\partial g}{\partial y}=0+4y-0-7xz (x\: and\: z\: held\: constant)}$$

$$\mathrm{\Rightarrow \frac{\partial g}{\partial y}=4y-7xz}$$

$$\mathrm{\frac{\partial g}{\partial z}=\frac{\partial }{\partial z}(5x^2+2y^2-6z^3-7xyz)}$$

$$\mathrm{\Rightarrow \frac{\partial g}{\partial z}=0+0-18z^2-7xy (x\: and\: y\: held\: constant)}$$

$$\mathrm{\Rightarrow \frac{\partial g}{\partial z}=-18z^2-7xy}$$

Example 2:

Find the double integral value of $\mathrm{∫ ∫(x^2 y+5x) dx\: dy.}$

Solution:

$$\mathrm{ Let\: I=\int[\int(x^2 y+5x) dx] dy}$$

$$\mathrm{\Rightarrow I=\int[\int(x^2 y+5x) dx] dy}$$

$$\mathrm{\Rightarrow I=\int[\frac{x^3 y}{3} +\frac{5x^2}{2}] dy}$$

$$\mathrm{\Rightarrow I=\frac{x^3 y^2}{6} +\frac{5x^3}{6}+c}$$

$\mathrm{\Rightarrow I=\frac{x^3 y^2+5x^3}{6}+c}$ (where c is an integrating constant)

Example 3:

Prove that if k is a constant, g(y,m)=sin (km) cos(y) is a solution of

$$\mathrm{\frac{\partial^2 g}{\partial m^2}=k^2\frac{\partial^2 g}{\partial y^2}}$$

Solution:

Let’s assume g(y,m)=sin (km) cos(y) is a solution of $\mathrm{\frac{\partial^2 g}{\partial m^2}=k^2\frac{\partial^2 g}{\partial y^2}}$. Hence, the function g must satisfy the above equation.

$$\mathrm{\frac{\partial g}{\partial m}=\frac{\partial }{\partial m}(sin (km) cos(y))=k cos(km) cos(y)}$$

Now,$\mathrm{\frac{\partial^2 g}{\partial m^2}=\frac{\partial }{\partial m}(k cos(km) cos(y))=-k^2 sin(km) cosy\dotso\dotso\dotso(1)}$

$$\mathrm{\frac{\partial g}{\partial m}=\frac{\partial }{\partial y}(sin (km) cos(y))=-sin(km) siny}$$

Now, $\mathrm{\frac{\partial^2 g}{\partial y^2}=\frac{\partial }{\partial y}(-sin(km) siny)=-sin(km) cosy}$

Multiplying k² to above expression, we will get

$$\mathrm{k^2\frac{\partial^2 g}{\partial y^2}=\frac{\partial }{\partial y}=-k^2 sin(km) cosy\dotso\dotso\dotso\dotso(2)}$$

From Eq. (1) and Eq. (2), we can conclude that g(y,m)=sin (km) cos(y) is a solution of $\mathrm{\frac{\partial^2 g}{\partial m^2}=k^2 \frac{\partial^2 g}{\partial y^2}}$.

Conclusion

The present tutorial gives a brief introduction about multivariable calculus and its various subtopics. The brief summaries of partial derivatives, integration, and partial differential equations have been stated in this tutorial. Moreover, some solved examples have been provided for better clarity of this concept. In conclusion, the present tutorial may be useful for understanding the basic concept of multivariable calculus.

FAQs

1.What do you mean by the order of a partial differential equation?

The order of the highest derivative term of the differential equation is known as the order of the partial differential equation. For example, consider a PDE

$\mathrm{\frac{\partial g}{\partial y}+\frac{\partial g}{\partial z}=yz-2.}$ The order of the highest derivative term is 1; the given PDE is called a first-order PDE.

2.What is the application of multivariable integration?

The double integral is commonly used to determine the surface area of a bounded region and the triple integral is used to evaluate the volume of an object.

3.What is the degree of a partial differential equation?

The degree of the highest derivative term of the differential equation is known as the degree of the partial differential equation. For example, consider a PDE

$\mathrm{\frac{\partial g}{\partial y}+\frac{\partial g}{\partial z}=yz-2.}$ The degree of the highest derivative term is 1.

4.What are the applications of multivariable calculus?

The applications of multivariable calculus include the designing and optimization of dynamic systems. In addition, it is applied for regression analysis in finance and engineering.

5.Can we determine the maxima and minima of a multiple variable functions?

Yes. We have to use the concept of multivariable calculus to determine the maxima and minima of a multiple variable functions.