IGNOU BSCFMT BMTC 102 Multivariable Calculus Solved Assignment English

Looking for the solved assignment for BMTC 102Multivariable Calculus? Here you can get the complete solved assignment in PDF format for your IGNOU course.

If you are preparing your assignment and need the answers in one place, this solved PDF can help you save time and understand how the questions can be answered. You can read it on your mobile, save it for later, or take a print if required.

BSCFMT 2026 - English

Before you order: Please check the course code, subject code, year and session with your assignment before purchasing.

BMTC 102: Multivariable Calculus

High Demand Verified Solution

★★★★★ 4.8/5 (32 Students)

₹80 ₹30

63% OFF You Save: ₹50

Language

Type

Choose Your Assignment (latest Selected) Check Question Included See Included Question

Title Name	IGNOU BSCFMT BMTC 102 Multivariable Calculus Solved Assignment English
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSCFMT
Course Name	Bachelor of Science (Mathematics)
Subject Code	BMTC 102
Subject Name	Multivariable Calculus
Year	2026
Session	-
Language	English
Assignment Code	BMTC 102/Assignment-1/2026
Product Description	Assignment of BSCFMT (Bachelor of Science (Mathematics)) 2026. Latest BMTC 102 2026 Solved Assignment Solutions
Format	Ready-to-Print PDF (.soft copy)

Need Help With This Assignment?

Looking for the solved assignment for this subject? You can get the complete solved PDF and use it while preparing your assignment.

✓ Complete Solved Assignment — Questions with their answers in one PDF.

✓ Save Your Time — No need to search different websites for every question.

✓ Easy to Read — Keep the PDF on your phone or take a print when required.

✓ Get It Online — Download your assignment after successful payment.

✓ Useful for Preparation — Read the answers and prepare your assignment with less hassle.

Just check once before buying: Make sure the Course Code, Subject Code, Year and Session shown above match your assignment.

📋 Assignment Content Preview

Please make sure your assignment questions match your current IGNOU session

BMTC 102 2026 - English

ASSIGNMENT

Course Code: BMTC-102

Assignment Code: BMTC-102/TMA/2026

Maximum Marks: 100

1. State whether the following statements are true or false. Give reasons for your answers.

(i) $\displaystyle \lim_{x \to 0} \frac{x^2 \sin \frac{1}{x}}{\sin x} = 1$

(ii) A real-valued function of three variables which is continuous everywhere is differentiable.

(iii) The function $F : \mathbb{R}^2 \to \mathbb{R}^2$ , defined by $F(x, y) = (y + 2, x + y)$ , is locally invertible at any $(x, y) \in \mathbb{R}^2$ .

(iv) $f : [-1,1] \times [-2,2] \to \mathbb{R}$ , defined by
$$f(x, y) = \begin{cases} x, & \text{if } y \text{ is rational} \\ 0, & \text{if } y \text{ is not rational} \end{cases}$
is integrable.

(v) The function $f : \mathbb{R}^2 \to \mathbb{R}$ , defined by $f(x, y) = 1 - y^2 + x^2$ , has an extremum at (0, 0).

2) (a) Find the following limits:

$\quad$ (i) $\displaystyle \lim_{x \to +\infty} \left( \frac{x^2}{8x^2 - 3} \right)^{1/3}$

$\quad$ (ii) $\displaystyle \lim_{x \to 0^+} (\sin x)^{\sin x}$

(b) Using only the definitions, find f_xy(0, 0) and f_yx(0, 0), if they exists, for the function

$$f(x, y) = \begin{cases} \frac{x^2 y}{\sqrt{x^2 + y^2}}, & (x, y) \neq (0, 0) \\ 0, & \text{otherwise} \end{cases}$

3) (a) Let the function f be defined by

$$f(x, y) = \begin{cases} \frac{3x^2 y^4}{x^4 + y^8}, & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases}$

$\quad$ Show that f has directional derivatives in all directions at (0, 0).

(b) Let $x = e^r \cos \theta, y = e^r \sin \theta$ and f be a continuously differentiable function of x and y, whose partial derivatives are also continuously differentiable. Show that

$$\frac{\partial^2 f}{\partial r^2} + \frac{\partial^2 f}{\partial \theta^2} = (x^2 + y^2) \left( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right)$

(c) Let $a = (1, 2, 3), b = (-5, 3, -2), c = (2, -4, 1)$ be three points in $\mathbb{R}^3$ .

Find |2b - a + 3c|.

4. (a) Find the centre of gravity of a thin sheet with density $\delta(x, y) = y$ , bounded by the curves $y = 4x^2$ and $x = 4$ .

(b) Find the mass of the solid bounded by $z = 1$ and $z = x^2 + y^2$ , the density function being $\delta(x, y, z) = |x|$ .

5. (a) State Green’s theorem, and apply it to evaluate

$$\int_C (3x^2 - 4y) \, dx - (2x + y^3) \, dy,$

Where C is the ellipse $4x^2 + 9y^2 = 36$ .

(b) Find the extreme values of the function

$f(x, y) = x^2 + y$ on the surface $x^2 + 2y^2 = 1$ .

6. (a) Check the continuity and differentiability of the function at (0,0) where

$$f(x, y) = \begin{cases} \frac{2x^3 y}{x^2 + y^2}, & (x, y) \neq (0, 0) \\ 0, & \text{otherwise} \end{cases}$

$\quad$ (b) Find the domain and range of the function f, defined by $\displaystyle f(x, y) = \frac{2xy}{x^2 + y^2}$ . Also find two level curves of this function. Give a rough sketch of them.

7. (a) Evaluate $\displaystyle \int_C (2x^2 + 3y^2) dx$ , where C is the curve given by

$$x(t) = at^2, y(t) = 2at, 0 \leq t \leq 1.$

$\quad$ (b) Use double integration of find the volume of the ellipsoid

$$\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{16} = 1.$

8. (a) Find the values of a and b, if

$$\lim_{x \to 0} \frac{x(1 + a \cos x) - b \sin x}{x^3} = 1$

$\quad$ (b) Suppose S and C are subsets of $\mathbb{R}^3$ . S is the unit open sphere with centre at the origin and C is the open cube $= \{P(x, y, z) \mid -1 < x < 1, -1 < y < 1, -1 < z < 1\}$ .

Which of the following is true. Justify your answer.

(i) $S \subset C$
(ii) $C \subset S$

(c) Identify the level curves of the following functions:
(i) $\sqrt{x^2 + y^2}$
(ii) $\sqrt{4 - x^2 - y^2}$
9. a) Using polar coordinates, show that $\displaystyle \lim_{\substack{x \to 0 \\ y \to 0}} \frac{x^3 - y^3}{x^2 + y^2} = 0$ . Also, find the two repeated limits.

$\quad$ b) Write $\displaystyle \int_0^1 \int_0^{\sqrt{1 - x^2}} (\sqrt{1 - y^2}) \, dy \, dx$ as an integral over a region D. Sketch the region D and show that it is of both types 1 and 2. Reverse the order of integration and evaluate it.

10. (a) Check if the following integrals are independent of path and evaluate those which are independent.

$$\text{i) } \int_{(0,0)}^{(3,4)} (6xy - y^3) \, dx + (3x^2 - 3xy^2) \, dy$

$$\text{ii) } \int_{(-1,4)}^{(3,8)} (3x^2 - 2y^2) \, dx - 4xy \, dy$

$\quad$ (b) Evaluate $\displaystyle \iiint_S z^2 \, dx \, dy \, dz$ , where S is the solid region between the spheres $\rho = 1$ and
$\quad$ $\rho = 2$ , by using spherical coordinates.

BMTC 102 2026 - Hindi

सत्रीय कार्य

पाठ्यक्रम कोड: BMTC-102
सत्रीय कार्य कोड: BMTC-102/TMA/2026
अधिकतम अंक: 100

1. बताइए निम्नलिखित कथन सत्य हैं या असत्य। अपने उत्तरों के कारण बताइए। (10)

(i) $\quad \lim_{x \to 0} \frac{x^2 \sin \frac{1}{x}}{\sin x} = 1$

(ii) $\quad$ तीन चरों वाला एक वास्तविक-मान फलन, जो सर्वत्र सतत है, अवकलनीय होता है।

(iii) $\quad F(x, y) = (y + 2, x + y)$ से परिभाषित फलन $F: \mathbf{R}^2 \to \mathbf{R}^2$ किसी भी बिन्दु $(x, y) \in \mathbf{R}^2$ पर स्थानिकतः व्युत्क्रमणीय होता है।

(iv) Image ignouacademy-com-ignou-bmtc-102-solved-assignment-html-p-ignou-57794
$\quad\quad$ से परिभाषित फलन $f: [-1,1] \times [-2,2] \to \mathbf{R}$ समाकलनीय होता है।

(v) $\quad f(x, y) = 1 - y^2 + x^2$ से परिभाषित फलन $f: \mathbf{R}^2 \to \mathbf{R}$ का (0, 0) पर एक चरम मान होता है।

2) (क) निम्नलिखित सीमा ज्ञात कीजिए :

(i) $\quad \lim_{x \to +\infty} \left( \frac{x^2}{8x^2 - 3} \right)^{1/3}$

(ii) $\quad \lim_{x \to 0^+} (\sin x)^{\sin x}$

(ख) केवल परिभाषाओं को लागू करके f_xy(0, 0) और f_yx(0, 0) ज्ञात कीजिए, जबकि फलन

Image ignouacademy-com-ignou-bmtc-102-solved-assignment-html-p-academy-50751

के लिए इनका अस्तित्व होता हो।

3) (क) मान लीजिए
$$f(x, y) = \begin{cases} \frac{3x^2 y^4}{x^4 + y^8}, & (x, y) \neq (0,0) \\ 0, & (x, y) = (0,0) \end{cases}$ $
दिखाइए कि (0,0) पर सभी दिशाओं में f दिक् अवकलज होते हैं।

(ख) मान लीजिए $x = e^r \cos \theta, y = e^r \sin \theta$ और f, x और y का एक संततः अवकलनीय फलन है जिसके आंशिक अवकलज भी संततः अवकलनीय हैं। दिखाइए कि
$$\frac{\partial^2 f}{\partial r^2} + \frac{\partial^2 f}{\partial \theta^2} = (x^2 + y^2) \left( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right)$ $

(ग) मान लीजिए $a = (1, 2, 3), b = (-5, 3, -2), c = (2, -4, 1), \mathbf{R}^3$ के तीन बिन्दु हैं।
|2b - a + 3c| ज्ञात कीजिए।

4. (क) वक्रों $y = 4x^{2}$ और $x = 4$ से परिबद्ध और $\delta(x,y) = y$ के घनत्व वाले एक पतली शीट का गुरुत्व केन्द्र ज्ञात कीजिए।

(ख) $z = 1$ और $z = x^{2} + y^{2}$ से परिबद्ध ठोस घनाकृति का द्रव्यमान ज्ञात कीजिए, जबकि घनत्व फलन, $\delta(x,y,z) = |x|$ हो।

5. (क) ग्रीन प्रमेय का कथन दीजिए और इसकी सहायता से
Image ignouacademy-com-ignou-bmtc-102-solved-assignment-html-p-multivariable-11689
जहाँ C, दीर्घवृत्त $4x^{2} + 9y^{2} = 36$ है।

(ख) पृष्ठ $x^{2} + 2y^{2} = 1$ पर फलन $f(x,y) = x^{2} + y$ के चरम मान ज्ञात कीजिए।

6. (क) (0,0) पर निम्नलिखित फलन f के सांतत्य और अवकलनीयता की जाँच कीजिए, जहाँ

Image ignouacademy-com-ignou-bmtc-102-solved-assignment-html-p-ignou-22445

(ख) फलन $f(x, y) = \frac{2xy}{x^2 + y^2}$ से परिभाषित फलन f का प्रांत और परिसर ज्ञात कीजिए। इस फलन के दो स्तर वक्र भी ज्ञात कीजिए। (4)

7. (क) $\int\limits_{C} (2x^2 + 3y^2) dx$ के मान निकालिए, जहाँ C $x(t) = at^2, y(t) = 2at, 0 \le t \le 1$ से प्राप्त वक्र है।

(ख) द्विशः समाकलन का प्रयोग करके दीर्घवृत्तज
$$\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{16} = 1$ $
का आयतन ज्ञात कीजिए।

8. (क) यदि $\lim_{x \to 0} \frac{x(1 + a \cos x) - b \sin x}{x^3} = 1$ तो a और b के मान ज्ञात कीजिए।

(ख) मान लीजिए कि S और C $\mathbf{R}^3$ के उपसमुच्चय हैं। S मूल–बिन्दु पर केन्द्र वाला एकक विवृत गोलक है तथा C विवृत घन $= \{P(x, y, z) \mid -1 < x < 1, -1 < y < 1, -1 < z < 1\}$ ।
निम्नलिखित में से कौनसा कथन सत्य है? अपने उत्तर की पुष्टि कीजिए।
(i) $S \subset C$
(ii) $C \subset S$

(ग) निम्नलिखित फलनों के स्तर वक्र ज्ञात कीजिए :
(i) $\sqrt{x^2 + y^2}$
(ii) $\sqrt{4 - x^2 - y^2}$

9. क) ध्रुवीय निर्देशांकों का प्रयोग करते हुए दिखाइए कि $\lim_{\substack{x \to 0 \\ y \to 0}} \frac{x^3 - y^3}{x^2 + y^2} = 0$ है। दो पुनरावृत्ति सीमाएँ ज्ञात कीजिए।
$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

ख) प्रदेश D पर एक समाकल के रूप में $\int\limits_{0}^{1} \int\limits_{0}^{\sqrt{1 - x^2}} \left( \sqrt{1 - y^2} \right) dy \, dx$ लिखिए। प्रदेश D का चित्र बनाइए और दिखाएं कि यह टाईप 1 और टाईप 2 दोनों है। समाकल के क्रम को उलटिए और इसका मूल्यांकन कीजिए।
$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

10. क) जाँच कीजिए कि निम्नलिखित समाकलन स्वतंत्र पथ है और जो स्वतंत्र है उनका मूल्यांकन कीजिए
i) $\int\limits_{(0,0)}^{(3,4)} (6xy - y^3) \, dx + (3x^2 - 3xy^2) \, dy$ .
ii) $\int\limits_{(-1,4)}^{(3,8)} (3x^2 - 2y^2) \, dx - 4xy \, dy$ .
$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

ख) निर्देशांकों का प्रयोग करते हुए $\iiint\limits_{S} z^2 dx \, dy \, dz$ का मूल्यांकन कीजिए जहाँ S गोले $\rho = 1$ और $\rho = 2$ के बीच ठोस प्रदेश है।
$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

❓ Frequently Asked Questions (FAQs)

Q: How will I receive the PDF?
A: Immediately after payment, the download link will appear.

Q: Is this hand-written or typed?
A: This is a professional typed computer PDF. You can use it as a reference for your handwritten submission.

➕Other Details

IGNOU BMTC 102 Solved Assignment 2026 – Prepare Smarter, Save Valuable Time

Searching for the 2026 IGNOU BMTC 102 Solved Assignment? Get a ready-to-use reference for BMTC 102 - Multivariable Calculus and make your assignment preparation simpler, faster and more organised. Our carefully prepared BMTC 102 Multivariable Calculus Solved Assignment is designed to help you understand the questions, study the important points and prepare your assignment answers with greater confidence.

Why spend hours searching through different sources when you can have your BMTC 102 solved assignment reference available in one convenient resource? Whether you are studying after work, preparing several IGNOU subjects together or simply want a dependable reference, this material can save you valuable preparation time.

BMTC 102 2026 Solved Assignment – What You Get

Latest 2026 assignment reference for BMTC 102 Multivariable Calculus
Question-wise answers arranged for easy study and reference
Clear and student-friendly presentation for easier understanding
English language version available
Digital soft copy / PDF for convenient access
Useful reference material for preparing your own assignment answers
Helps you identify important points, answer structure and relevant content

Stop Struggling With Every Question. Start Preparing With a Better Reference.

Assignment preparation can become time-consuming when you have multiple subjects, limited study hours and a long list of questions to complete. A good reference can help you move from “What should I write?” to “Now I understand how to prepare this answer.”

The IGNOU BMTC 102 Solved Assignment 2026 gives you a convenient study reference so you can review the questions, understand the expected subject matter and organise your own answers more efficiently.

Study less randomly. Prepare more systematically. Keep the reference with you while working on your assignment and use it whenever you need help understanding a question or arranging the important points.

Get Your BMTC 102 Multivariable Calculus Solved Assignment 2026

Course: Bachelor of Science (Mathematics) (BSCFMT)
Subject: BMTC 102 - Multivariable Calculus
Session: January 2026 - December 2026
Language: English

Need the Help Book / Solved Assignment?
Call us: 9199852182 / 9852900088
WhatsApp: 9199852182

Your Time Matters – Make Assignment Preparation Easier

If you are balancing IGNOU studies, work, family responsibilities or other subjects, every hour of study matters. Having a structured reference beside you can make it easier to understand the assignment and focus your time on preparing your own submission.

Instead of repeatedly searching for answers across different websites and sources, keep your BMTC 102 Multivariable Calculus Solved Assignment 2026 ready for reference whenever you study.

A Useful Reference for Your IGNOU Assignment Preparation

Understand the question → Find the relevant points → Study the answer → Prepare your own assignment with confidence.

IGNOU BSCFMT Assignments 2026

The BSCFMT (Bachelor of Science (Mathematics)) programme includes assignment work as part of the academic requirements applicable to students. Students should always verify the latest assignment questions, instructions and submission requirements from the official IGNOU website.

Our BMTC 102 Solved Assignment is independently prepared reference material intended to assist students with understanding and preparing their answers. It is not an official IGNOU publication.

Why Keep a Solved Assignment Reference?

A useful reference is more than simply a collection of answers. It can help you see how a question can be approached, what important points need attention and how the subject matter can be organised into a clear answer.

Save preparation time with organised reference material
Understand difficult questions more easily
Review important concepts before writing your answers
Improve answer organisation with a structured reference
Study conveniently whenever and wherever you need it

Product Information

Course Name	Bachelor of Science (Mathematics)
Course Code	BSCFMT
Subject Code	BMTC 102
Subject	Multivariable Calculus
Programme	BACHELOR DEGREE PROGRAMMES Courses
Language	English
Material Type	Solved Assignment – Soft Copy / PDF

Ready to Make Your BMTC 102 Assignment Preparation Easier?

Get the 2026 BMTC 102 Multivariable Calculus Solved Assignment as a convenient reference for your assignment preparation.

Call: 9199852182 / 9852900088
WhatsApp: 9199852182

IGNOU BMTC 102 2026 Solved Assignment
Get your reference copy and start preparing with confidence.

Important Note for IGNOU Students

Always check the latest assignment questions and submission instructions on the official IGNOU website before preparing and submitting your assignment. Use this solved assignment as a study and reference resource and prepare your own submission according to the applicable IGNOU requirements.

Why Choose IGNOU Academy for Your Assignments?

Getting your assignments right is an important part of your IGNOU academic journey.IGNOU Academy provides carefully prepared reference materials designed to makeassignment preparation simpler, clearer and more convenient for students.

Latest Curriculum: Materials are prepared with the current IGNOU syllabus and assignment requirements in mind.
Proper Assignment Format: Understand the structure, presentation and answer style generally expected in an IGNOU assignment.
Simple Explanations: Difficult topics are explained in clear and easy-to-understand language.
Exam Preparation: Assignment material can also be useful for revising important concepts before the Term-End Examination.
Student-Friendly Language: Clear English/Hindi content helps students understand topics without unnecessary complexity.
Trusted by IGNOU Learners: Students across India use IGNOU Academy as a convenient source of assignment and study-reference material.

These materials are provided as reference study guides. Students should understandthe concepts, follow IGNOU guidelines and prepare their own original assignmentswhere required.

How to Get Your Solved Assignment PDF

Visit IGNOU Academy:Go to www.ignouacademy.com.
Find Your Course:Search for your programme, course code and subject.
Select the Session:Choose the latest available reference material for your academic session.
Complete Your Order:Add the required material to your cart, log in or register, and complete the payment.
Access Your Material:After successful payment, access your purchased study material from your account.

Step-by-Step: Finding IGNOU Assignment Question Papers

Open the Website:Visit www.ignouacademy.com.
Open Assignment Question Papers:Select the relevant IGNOU assignment question paper section.
Filter Your Requirement:Select your course, session and medium such as English or Hindi.
Open the Paper:View or download the required question paper for your preparation.

How to Submit Your IGNOU Assignments

Prepare Neatly:Use the format and submission instructions applicable to your programme and study centre.
Complete the Front Page:Enter your Name, Enrollment Number, Programme/Course Code, Subject and Study Centre details correctly.
Offline Submission:If your study centre requires physical submission, submit the assignment as instructed and retain the acknowledgement.
Online Submission:Where online submission is permitted, follow the instructions issued by your Regional Centre or Study Centre. Keep proof of submission for your records.

Tracking Your Assignment Submission Status

After submitting your assignments, students can check whether their assignmentmarks and submission details have been updated through the applicable official IGNOU system.

Open the Official IGNOU Student Area:Visit the official IGNOU website and access the relevant student services.
Find Assignment Status:Open the assignment-status service applicable to your programme.
Enter Your Details:Provide your Enrollment Number and other requested programme details.
Check the Status:Review the available submission, evaluation and marks information.

A Quick Tip for IGNOU Students

Assignments are an important part of the IGNOU evaluation process. Preparing themcarefully, understanding the questions, following the prescribed format and submittingthem within the applicable deadline can help you stay on track academically.

Smart Study + Proper Preparation + Timely Submission = Better Academic Planning

Need Help? Contact IGNOU Academy WhatsApp: +91 9199852182 | Website: www.ignouacademy.com

IGNOU Solved Assignments Assignments Ignou Projects Project Previous Year Solved Exam Papers Notes Notes Sample Question Papers Sample Question Papers