🔔 BIG UPDATE FOR GATE ASPIRANTS 🔔📘 GATE Engineering Mathematics – Vector Calculus Masterclass🎓 Gradient | Divergence | Curl | Line Integrals | Green’s Theorem
🔔 BIG UPDATE FOR GATE ASPIRANTS 🔔📘 GATE Engineering Mathematics – Vector Calculus Masterclass🎓 Gradient Divergence Curl Line Integrals Green’s Theorem
🔔 BIG UPDATE FOR GATE ASPIRANTS 🔔
📘 GATE Engineering Mathematics – Vector Calculus Masterclass
🎓 Gradient | Divergence | Curl | Line Integrals | Green’s Theorem
🔥 By Grandmaster Bikram Sutradhar
🏆 5× World Record Holder | Best Teacher Awardee 2023
🎯 IIT JAM AIR 392 | Founder – bAstronautWay
🌐 Website: sirbikramsutradhar.com
📞 Contact: 9863002294
📚 FORMULAS & CONCEPTS – MADE EASY ✅
1️⃣ Gradient (grad f)
Formula: grad f = (df/dx, df/dy, df/dz)
📌 Always points perpendicular to level surfaces.
2️⃣ Divergence (div F)
Formula: div F
🔁 Measures how much a vector field spreads out.
📌 If div F = 0 → field is incompressible.
3️⃣ Curl (curl F)
Formula: curl F = ( ∂Fz/∂y – ∂Fy/∂z, ∂Fx/∂z – ∂Fz/∂x, ∂Fy/∂x – ∂Fx/∂y )
🔃 Measures rotation in a vector field.
📌 If curl = 0 → field is irrotational.
4️⃣ Line Integral
Formula: ∫ F · dr
🧰 Used to calculate work done by a force field along a path.
📌 If F = grad f → integral depends only on end points (not the path).
5️⃣ Green’s Theorem (2D only)
Formula: ∮ (P dx + Q dy) = ∬ (∂Q/∂x – ∂P/∂y) dxdy
🔄 Connects line integrals around a closed curve to area integrals.
📌 Useful for converting boundary problems into area-based ones.
🎯 50 MOST IMPORTANT QUESTIONS (For GATE + JAM + Core Concepts)
✅ Concept-based | 💡 Problem-solving | 🔥 GATE trend focused
- Find grad f when f = x² + y² + z²
- Prove that curl(grad f) = 0 for any function f
- Find divergence of F = x i + y j + z k
- Show that the curl of a radial vector is 0
- State and explain Green’s Theorem
- Evaluate line integral of F = y i + x j along a circle
- Is a conservative field path-dependent?
- Prove that div(curl F) = 0 always
- What is the physical meaning of gradient?
- Derive Laplacian (div of grad) of a scalar field
…
📩 Get full 50-question PDF by messaging “GATE Vector 50” on WhatsApp: 9863002294
🚀 Why Learn from Sir Bikram Sutradhar?
✨ 5× World Record Holder
🎯 IIT JAM AIR 392
🏆 Best Teacher Awardee 2023
🧠 Known for visual, easy, and deep concept-based teaching
🌍 Trusted by 2.5L+ students worldwide
📞 Join Now: 9863002294
🌐 Visit: sirbikramsutradhar.com
📥 DM “VECTOR MASTERCLASS” to enroll today!
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Written By
Full Stack Developer and 5-Time World Record Holder, Grandmaster Bikram Sutradhar
bAstronautWay : A Government-Approved Trademark Brand
SirBikramSutradhar on YouTube
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Written By
Full Stack Developer and 5-Time World Record Holder, Grandmaster Bikram Sutradhar
bAstronautWay
SirBikramSutradhar on YouTube
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ICSE CLASS 10 ICSE CLASS 10 BIOLOGY BASTRONAUTWAY SirBikramSutradhar Bikram Sutradhar GrandMaster Bikram Sutradhar selina biology solutions ICSE Biology Selina Solution
📚 Class 9 Science – Complete Table of Contents (विषय-सूची)
CBSE | NCERT Curriculum | Bilingual (English + Hindi)
By Grandmaster Bikram Sutradhar | SirBikramSutradhar | BASTRONAUTWAY
🔬 PHYSICS (भौतिक विज्ञान)
| 🔢 | Chapter Title (English) | अध्याय का नाम (हिंदी) | Page No. |
| 1️⃣ | Motion | गति | |
| 2️⃣ | Force and Laws of Motion | बल तथा गति के नियम | |
| 3️⃣ | Gravitation | गुरुत्वाकर्षण | |
| 4️⃣ | Work and Energy | कार्य तथा ऊर्जा | |
| 5️⃣ | Sound | ध्वनि |
🧪 CHEMISTRY (रसायन विज्ञान)
| 🔢 | Chapter Title (English) | अध्याय का नाम (हिंदी) | Page No. |
| 1️⃣ | Matter in Our Surroundings | हमारे आस-पास के पदार्थ | |
| 2️⃣ | Is Matter Around Us Pure? | क्या हमारे आस-पास के पदार्थ शुद्ध हैं? | |
| 3️⃣ | Atoms and Molecules | परमाणु एवं अणु | |
| 4️⃣ | Structure of the Atom | परमाणु की संरचना |
🧬 BIOLOGY (जीव विज्ञान)
| 🔢 | Chapter Title (English) | अध्याय का नाम (हिंदी) | Page No. |
| 1️⃣ | The Fundamental Unit of Life | जीवन की मौलिक इकाई | |
| 2️⃣ | Tissues | ऊतक | |
| 3️⃣ | Improvement in Food Resources | धनों में सुधार |
📘 ADDITIONAL SECTIONS (अतिरिक्त अनुभाग)
| 📖 Section | Title (English + Hindi) |
| 🔹 | Glossary / पारिभाषिक शब्दावली |
| 🔹 | Answer Keys / उत्तरमाला |
| 🔹 | Summary / सारांश |
| 🔹 | Important Diagrams / महत्वपूर्ण चित्र |
| 🔹 | NCERT Exercises / एनसीईआरटी अभ्यास |
| 🔹 | Sample Papers & MCQs / मॉडल पेपर व वस्तुनिष्ठ प्रश्न |
✅ All content follows the official NCERT Class 9 Science textbook structure. Each chapter will include explanations, bilingual notes, definitions, diagrams, MCQs, FAQs, and practice exercises under the BASTRONAUTWAY learning system.
class 10 Maths ncert solution.
CBSE Class 10 Mathematics (2025) NCERT Syllabus:
CBSE Class 10 Mathematics NCERT Syllabus (2025) – Chapter-Wise List
CBSE Class 10 Mathematics NCERT Syllabus (2025) – Chapter-Wise List
chapter 4. Quadratic Equations
chapter 5. Arithmetic Progressions
chapter 6. Triangles
chapter 7. Coordinate Geometry
chapter 8. Introduction to Trigonometry
chapter 9. Some Applications of Trigonometry
chapter 11. Areas Related to Circles
chapter 12. Surface Areas and Volumes
CBSE Class 10 Science (2025) NCERT Syllabus:
CBSE Class 10 Science NCERT Syllabus (2025) – Chapter-Wise List
CBSE Class 10 Science NCERT Syllabus (2025) – Chapter-Wise List
Chapter 1 Chemical Reactions and Equations
Chapter 2 Acids, Bases and Salts
Chapter 3 Metals and Non-metals
Chapter 4 Carbon and its Compounds
Chapter 5 Life Processes
Chapter 6 Control and Coordination
Chapter 7 How do Organisms Reproduce?
Chapter 8 Heredity
Chapter 9 Light – Reflection and Refraction
Chapter 10 The Human Eye and the Colourful World
Chapter 11 Electricity
Chapter 12 Magnetic Effects of Electric Current
Chapter 13 Our Environment
📖 Chapter 12 – Organic Chemistry (Selina Textbook)
Table of Contents:
- 12A. Organic Compounds
- 12B. Hydrocarbons: Alkanes
- 12C. Hydrocarbons: Alkenes
- 12D. Hydrocarbons: Alkynes
- 12E. Alcohols
- 12F. Carboxylic Acids
- Exercise 12 MISCELLANEOUS
- Glossary
- Model Question Paper –
📌 Exercise 12 MISCELLANEOUS – Focus Areas:
- 🧬 Consolidated practice from the entire Organic Chemistry chapter
- 🧪 Application-based questions covering alkanes, alkenes, alkynes, alcohols, and carboxylic acids
- 🧾 Writing structural formulas and chemical equations
- 💡 Distinguishing reactions, isomer identification, and conversions
- 📘 Full-syllabus revision to boost conceptual clarity and board exam readiness
Each question in this section is designed to:
- ✅ Strengthen cross-topic understanding
- 📝 Prepare students for application-based and reasoning-based questions
- 🧠 Improve problem-solving speed and exam performance
class 5
📚 Class 9 Science – Complete Table of Contents (विषय-सूची)
CBSE | NCERT Curriculum | Bilingual (English + Hindi)
By Grandmaster Bikram Sutradhar | SirBikramSutradhar | BASTRONAUTWAY
🔬 PHYSICS (भौतिक विज्ञान)
| 🔢 | Chapter Title (English) | अध्याय का नाम (हिंदी) | Page No. |
| 1️⃣ | Motion | गति | |
| 2️⃣ | Force and Laws of Motion | बल तथा गति के नियम | |
| 3️⃣ | Gravitation | गुरुत्वाकर्षण | |
| 4️⃣ | Work and Energy | कार्य तथा ऊर्जा | |
| 5️⃣ | Sound | ध्वनि |
🧪 CHEMISTRY (रसायन विज्ञान)
| 🔢 | Chapter Title (English) | अध्याय का नाम (हिंदी) | Page No. |
| 1️⃣ | Matter in Our Surroundings | हमारे आस-पास के पदार्थ | |
| 2️⃣ | Is Matter Around Us Pure? | क्या हमारे आस-पास के पदार्थ शुद्ध हैं? | |
| 3️⃣ | Atoms and Molecules | परमाणु एवं अणु | |
| 4️⃣ | Structure of the Atom | परमाणु की संरचना |
🧬 BIOLOGY (जीव विज्ञान)
| 🔢 | Chapter Title (English) | अध्याय का नाम (हिंदी) | Page No. |
| 1️⃣ | The Fundamental Unit of Life | जीवन की मौलिक इकाई | |
| 2️⃣ | Tissues | ऊतक | |
| 3️⃣ | Improvement in Food Resources | धनों में सुधार |
📘 ADDITIONAL SECTIONS (अतिरिक्त अनुभाग)
| 📖 Section | Title (English + Hindi) |
| 🔹 | Glossary / पारिभाषिक शब्दावली |
| 🔹 | Answer Keys / उत्तरमाला |
| 🔹 | Summary / सारांश |
| 🔹 | Important Diagrams / महत्वपूर्ण चित्र |
| 🔹 | NCERT Exercises / एनसीईआरटी अभ्यास |
| 🔹 | Sample Papers & MCQs / मॉडल पेपर व वस्तुनिष्ठ प्रश्न |
✅ All content follows the official NCERT Class 9 Science textbook structure. Each chapter will include explanations, bilingual notes, definitions, diagrams, MCQs, FAQs, and practice exercises under the BASTRONAUTWAY learning system. class
